THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


JOHN  S. 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 

Cyclopedia 

of 

Civil   Engineering 


A  General  Reference  Work 

ON   SURVEYING,   RAILROAD   ENGINEERING,  STRUCTURAL   ENGINEERING,   ROOFS 

AND   BRIDGES,    MASONRY   AND   REINFORCED   CONCRETE,    HIGHWAY 

CONSTRUCTION,    HYDRAULIC   ENGINEERING,    IRRIGATION. 

RIVER   AND   HARBOR   IMPROVEMENT,    MUNICIPAL 

ENGINEERING,    COST  ANALYSIS,    ETC. 


Editor-in-  Chief 
FREDERICK  E.  TURNEAURE,  C.  E.,  Dr.  Eng. 

DEAN,    COLLEGE   OF  ENGINEERING,    UNIVERSITY   OF   WISCONSIN 


Assisted  by  a  Corps  of 

CIVIL  AND  CONSULTING  ENGINEERS   AND  TECHNICAL   EXPERTS   OF  THK 
HIGHEST   PROFESSIONAL   STANDING 


Illustrated  with  over  Three  Thousand  Engravings 


EIGHT    VOLUMES 


CHICAGO 

AMERICAN   TECHNICAL  SOCIETY 
1908 


COPYRIGHT, 


AMERICAN  SCHOOL  OF  CORRESPONDENCE 


COPYRIGHT,    1908 
BY 

AMERICAN  TECHNICAL  SOCIETY 


Entered  at  Stationers'  Hall,  London. 
All  Rights  Reserved. 


Uhrary 

n 
i+t 


. 

Editor-in-Chief 
FREDERICK  E.  TURNEAURE,  C.  E.,  Dr.  Eng. 

Dean,  College  of  Engineering,  University  of  Wisconsin 


Authors   and   Collaborators 

WALTER  LORING  WEBB,  C.  E. 

Consulting  Civil  Engineer 

American  Society  of  Civil  Engineers 

Author  of  "Railroad  Construction,  '  "Economics  of  Railroad  Construction,"  etc. 


FRANK  O.  DUFOUR,  C.  E. 

Assistant  Professor  of  Structural  Engineering,  University  of  Illinoi 
American  Society  of  Civil  Engineers 
American  Society  for  Testing  Materials 


HALBERT  P.  GILLETTE,  C.  E. 

Consulting  Engineer 
American  Society  of  Civil  Engineers 
Managing  Editor  "Engineering-Contracting" 

Author  of  "Handbook  of  Cost  Data  for  Contractors  and  Engineers,"  "Earthwork 
and  its  Cost,"  "Rock  Excavation—  Methods  and  Cost" 


ADOLPH  BLACK,  C.  E. 

Adjunct  Professor  of  Civil  Engineering,  Columbia  University,  N.  Y. 


EDWARD  R.  MAURER,  B.  C.  E. 

Professor  of  Mechanics,  University  of  Wisconsin 

Joint  Author  of  "Principles  of  Reinforced  Concrete  Construction" 


W.  HERBERT  GIBSON,  B.  S.,  C.  E. 

Civil  Engineer 

Designer  of  Reinforced  Concrete 

V 

AUSTIN  T.  BYRNE 

Civil  Engineer 

Author  of  "Highway  Construction,"  "Materials  and  Workmanship 


Authors  and  Collaborators— Continued 


FREDERICK  E.  TURNEAURE,  C.  E.,  Dr.  Eng. 

Dean  of  the  College  of  Engineering,  and  Professor  of  Engineering,  University  of 

Wisconsin 

American  Society  of  Civil  Engineers 
Joint  Author  of  "Principles  of  Reinforced  Concrete  Construction,"  "Public  Water 

Supplies,"  etc. 


THOMAS  E.  DIAL,  B.  S. 

Instructor  in  Civil  Engineering,  American  School  of  Correspondence 

Formerly  with  Engineering  Department,  Atchison,  Topeka  &  Santa  Fe  Railroad 


ALFRED  E.  PHILLIPS,  C.  E.,  Ph.  D. 

Head  of  Department  of  Civil  Engineering,  Armour  Institute  of  Technology 
^» 

DARWIN  S.  HATCH,  B.  S. 

Instructor  in  Mechanical  Engineering,  American  School  of  Correspondence 


CHARLES  E.  MORRISON,  C.  E.,  A.  M. 

Instructor  in  Civil  Engineering,  Columbia  University,  N.  Y. 
Author  of  "Highway  Engineering." 


ERVIN  KENISON,  S.  B. 

Instructor  in  Mechanical  Drawing-,  Massachusetts  Institute  of  Technology 
*f 

EDWARD  B.  WAITE 

Head  of  Instruction  Department,  American  School  of  Correspondence 
American  Society  of  Mechanical  Engineers 
Western  Society  of  Engineers 

^« 

EDWARD  A.  TUCKER,  S.  B. 

Architectural  Engineer 

American  Society  of  Civil  Engineers 

<y» 

ERNEST  L.  WALLACE,  S.  B. 

Instructor  in  Electrical  Engineering,  American  School  of  Correspondence 
American  Institute  of  Electrical  Engineers 


A.  MARSTON,  C.  E. 

Dean  of  Division  of  Engineering  and  Professor  of  Civil  Engineering,  Iowa  State 

College 

American  Society  of  Civil  Engineers 
Western  Society  of  Civil  Engineers 


Authors  and  Collaborators-Continued 


CHARLES  B.  BALL 

Civil  and  Sanitary  Engineer 

Chief  Sanitary  Inspector,  City  of  Chicago 

American  Society  of  Civil  Engineers 


ALFRED  E.  ZAPF,  S.  B. 

Secretary.  American  School  of  Correspondence 
^« 

SIDNEY  T.  STRICKLAND,  S.  B. 

Massachusetts  Institute  of  Technology 
Ecole  des  Beaux  Arts,  Paris 

RICHARD  T.  DANA 

Consulting  Engineer 
American  Society  of  Civil  Engineers 
Chief  Engineer,  Construction  Service  Co. 
•V* 

ALFRED  S.  JOHNSON,  A.  M.,  Ph.  D. 

Textbook  Department,  American  School  of  Correspondence 
Formerly  Instructor,  Cornell  University 
Royal  Astronomical  Society  of  Canada 

WILLIAM  BEALL  GRAY 

Sanitary  Engineer 

National  Association  of  Master  Plumbers 

United  Association  of  Journeyman  Plumbers 

V 

R.  T.  MILLER,  Jr.,  A.  M.,  LL.  B. 

President  American  School  of  Correspondence 


GEORGE  R.  METCALFE,  M.  E. 

Head  of  Technical  Publication  Department,  Westinghouse  Electric  &  Manufac- 
turing Co. 

Formerly  Technical  Editor,  "Street- Rail  way  Review" 
Formerly  Editor  "The  Technical  World  Magazine" 


MAURICE  LE  BOSQUET,  S.  B. 

Massachusetts  Institute  of  Technology 

British  Society  of  Chemical  Industry,  American  Chemical  Society,  etc. 


HARRIS  C.  TROW,  S.  B.,  Managing  Editor 

Editor  of  Textbook  Department,  American  School  of  Correspondence 
American  Institute  of  Electrical  Engineers 


Authorities    Consulted 


THE  editors  have  freely  consulted  the  standard  technical  literature  of 
America  and  Europe  in  the  preparation  of  these  volumes.    They  desire 
to  express  their  indebtedness,  particularly,  to  the  following  eminent 
authorities,  whose  well-known  treatises  should  be  in  the  library  of  everyone 
interested  in  Civil  Engineering. 

Grateful  acknowledgment  is  here  made  also  for  the  invaluable  co-opera- 
tion of  the  foremost  Civil,  Structural,  Railroad,  Hydraulic,  and  Sanitary 
Engineers  in  making  these  volumes  thoroughly  representative  of  the  very 
best  and  latest  practice  in  every  branch  of  the  broad  field  of  Civil  Engineer- 
ing; also  for  the  valuable  drawings  and  data,  illustrations,  suggestions, 
criticisms,  and  other  courtesies. 


WILLIAM  G.  RAYMOND,  C.  E. 

Dean  of  the  School  of  Applied  Science  and  Professor  of  Civil  Engineering  in  the  State 

University  of  Iowa;  American  Society  of  Civil  Engineers. 

Author  of  "A  Textbook  of  Plane  Surveying,"  "The  Elements  of  Railroad  Engineering." 
^" 

JOSEPH  P.  FRIZELL 

Hydraulic  Engineer  and  Water-Power  Expert;  American  Society  of  Civil  Engineers. 
Author  of  "  Water  Power,  the  Development  and  Application  of  the  Energy  of  Flowing 
Water."  ^. 

FREDERICK  E.  TURNEAURE,  C.  E.,  Dr.  Eng. 

Dean  of  the  College  of  Engineering  and  Professor  of  Engineering,  University  of  Wisconsin. 
Joint  Author  of  "Public  Water  Supplies,"  "Theory  and  Practice  of  Modern  Framed 
Structures."  "  Principles  of  Reinforced  Concrete  Construction." 
** 

H.  N.  OGDEN,  C.  E. 

Assistant  Professor  of  Civil  Engineering,  Cornell  University. 
Author  of  "Sewer  Design." 

V 

DANIEL  CARHART,  C.  E. 

Professor  of  Civil  Engineering  in  the  Western  University  of  Pennsylvania. 
Author  of  "A  Treatise  on  Plane  Surveying." 
^ 

HALBERT  P.  GILLETTE 

Editor  of  Engineering- Contracting;  American  Society  of  Civil  Engineers;  Late  Chief 

Engineer,  Washington  State  Railroad  Commission. 
Author  of  "  Handbook  of  Cost  Data  for  Contractors  and  Engineers." 


CHARLES  E.  GREENE,  A.  M.,  C.  E. 

Late  Professor  of  Civil  Engineering,  University  of  Michigan. 

Author  of  "Trusses  and  Arches,  Graphic  Method,"  "Structural  Mechanics.' 


Authorities  Consulted— Continued 


A.  PRESCOTT  FOLWELL 

Editor  of  Municipal  Journal  and  Engineer;  Formerly  Professor  of  Municipal  Engineer- 

ing, Lafayette  College. 
Author  of  "Water  Supply  Engineering,"  "Sewerage." 


LEVESON  FRANCIS  VERNON-HARCOURT,  M.  A. 

Emeritus  Professor  of  Civil  Engineering  and  Surveying,  University  College,  London; 

Institution  of  Civil  Engineers. 
Author  of  "Rivers  and  Canals,"  "Harbors  and  Docks,"  "Achievements  in  Engineer- 

ing," "Civil  Engineering  as  Applied  in  Construction." 


PAUL  C.  NUGENT,  A.  M.,  C.  E. 

Professor  of  Civil  Engineering,  Syracuse  University. 
Author  of  "  Plane  Surveying." 

^* 

FRANK  W.  SKINNER 

Consulting  Engineer;  Associate  Editor  of  The  Engineering  Record;  Non-Resident  Lec- 

turer on  Field  Engineering  in  Cornell  University. 
Author  of  "  Types  and  Details  of  Bridge  Construction." 


HANBURY  BROWN,  K.  C.  M.  G. 

Member  of  the  Institution  of  Civil  Engineers. 
Author  of  "Irrigation,  Its  Principles  and  Practice." 


SANFORD  E.  THOMPSON,  S.  B.,  C.  E. 

American  Society  of  Civil  Engineers. 

Joint  Author  of  "A  Treatise  on  Concrete,  Plain  and  Reinforced." 


JOSEPH  KENDALL  FREITAG,  B.  S.,  C.  E. 

American  Society  of  Civil  Engineers. 

Author  of  "Architectural  Engineering,"  "  Fireproofing  of  Steel  Buildings." 
**• 

AUSTIN  T.  BYRNE,  C.  E. 

Civil  Engineer. 

Author  of  "Highway  Construction,"  "Inspection  of  Materials  and  Workmanship  Em- 
ployed in  Construction."  ~ 

JOHN  F.  HAYFORD,  C.  E. 

Inspector  of  Geodetic  Work  and  Chief  of  Computing  Division,  Coast  and  Geodetic  Survey; 

American  Society  of  Civil  Engineers. 
Author  of  "A  Textbook  of  Geodetic  Astronomy." 


WALTER  LORING  WEBB,  C.  E. 

Consulting  Civil  Engineer;  American  Society  of  Civil  Engineers. 

Author  of  "Railroad  Construction   in  Theory  and  Practice,"  "Economics  of  Railroad 
Construction,"  etc. 


Authorities  Consulted— Continued 


EDWARD  R.  MAURER,  B.  C.  E. 

Professor  of  Mechanics,  University  of  Wisconsin. 
Joint  Author  of  "Principles  of  Reinforced  Concrete  Construction." 
V 

HERBERT  M.  WILSON,  C.  E. 

Geographer  and  Former  Irrigation  Engineer,  United  States  Geological  Survey:  American 

Society  of  Civil  Engineers. 
Author  of  "  Topographic  Surveying,"  "  Irrigation  Engineering,"  etc. 


MANSFIELD  MERRIMAN,  C.  E.,  Ph.  D. 

Professor  of  Civil  Engineering,  Lehigh  University. 

Author  of  "  The  Elements  of  Precise  Surveying  and  Geodesy,"  "A  Treatise  on  Hydraul- 
ics," "Mechanics  of  Materials,"  "  Retaining  Walls  and  Masonry  Dams,"  "Introduction 
to  Geodetic  Surveying,"  "A  Textbook  on  Roofs  and  Bridges,"  "A  Handbook  for 
Surveyors,1'  etc. 

DAVID  M.  STAUFFER 

American  Society  of  Civil  Engineers;   Institution  of  Civil  Engineers;   Vice-President, 

Engineering  News  Publishing  Co. 
Author  of  '  Modern  Tunnel  Practice." 

^» 

CHARLES  L.  CRANDALL 

Professor  of  Railroad  Engineering  and  Geodesy  in  Cornell  University. 
Author  of  "A  Textbook  on  Geodesy  and  Least  Squares." 


N.  CLIFFORD  RICKER,  M.  Arch. 

Professor  of  Architecture,  University  of  Illinois;  Fellow  of  the  American  Institute  of 

Architects  and  of  the  Western  Association  of  Architects. 

Author  of  "  Elementary  Graphic  Statics  and  the  Construction  of  Trussed  Roofs." 
*• 

JOHN  C.  TRAUTWINE 

Civil  Engineer. 

Author  of  "The  Civil  Engineer's  Pocketbook." 
^r« 

HENRY  T.  BOVEY 

Professor  of  Civil  Engineering  and  Applied  Mechanics,  McGill  University,  Montreal. 
Author  of  "A  Treatise  on  Hydraulics." 


WILLIAM  H.  BIRKMIRE,  C.  E. 

Author  of  "Planning  and  Construction  of  High  Office  Buildings,"  "Architectural  Iron 
and  Steel,  and  Its  Application  in  the  Construction  of  Buildings,"  "Compound  Riv- 
eted Girders,"  "  Skeleton  Structures,'  etc. 
^ 

IRA  O.  BAKER,  C.  E. 

Professor  of  Civil  Engineering,  University  of  Illinois. 

Author  of  "A  Treatise  on  Masonry  Construction,"  "  Engineers'  Surveying  Instruments, 
Their  Construction,  Adjustment,  and  Use,"  "Roads  and  Pavements." 


Authorities  Consulted— Continued 


JOHN  CLAYTON  TRACY,  C.  E. 

Assistant  Professor  of  Structural  Engineering,  Sheffield  Scientific  School,  Yale  University. 
Author  of  "  Plane  Surveying:  A  Textbook  and  Pocket  Manual." 


FREDERICK  W.  TAYLOR,  M.  E. 

Joint  Author  of  "A  Treatise  on  Concrete,  Plain  and  Reinforced." 

*• 

JAMES  J.  LAWLER 

Author  of  "Modern  Plumbing,  Steam  and  Hot- Water  Heading." 


FRANK  E.  KIDDER,  C.  E.,  Ph.  D. 

Consulting  Architect  and  Structural  Engineer;   Fellow  of  the  American  Institute  of  . 

Architects. 
Author  of  "Architect's  and  Builder's  Pocketbook,"  "  Building  Construction  and  Super- 

intendence, Part  I,  Masons'  Work;  Part  II,  Carpenters'  Work;  Part  III,  Trussed  Roofs 

and  Roof  Trusses,"  "Strength  of  Beams,  Floors,  and  Roofs." 
^« 

WILLIAM  H.  BURR,  C.  E. 

Professor  of  Civil  Engineering,  Columbia  University;  Consulting  Engineer;   American 

Society  of  Civil  Engineers;  Institution  of  Civil  Engineers. 
Author  of  "  Elasticity  and  Resistance  of  the  Materials  of  Engineering;"  Joint  Author  of 

"  The  Design  and  Construction  of  Metallic  Bridges." 


WILLIAM  M.  GILLESPIE,  LL.  D. 

Formerly  Professor  of  Civil  Engineering  in  Union  University. 
Author  of  "  Land  Surveying  and  Direct  Leveling,"  "  Higher  Surveying." 
^« 

GEORGE  W.  TILLSON,  C.  E. 

President  of  the  Brooklyn  Engineers'  Club;  American  Society  of  Civil  Engineers;  Ameri- 
can Society  of  Municipal  Improvements;  Principal  Assistant  Engineer,  Department 
of  Highways,  Brooklyn. 

Author  of  "  Street  Pavements  and  Street  Paving  Material." 
^- 

G.  E.  FOWLER 

Civil  Engineer;  President,  The  Pacific  Northwestern  Society  of  Engineers;    American 

Society  of  Civil  Engineers. 
Author  of  "  Ordinary  Foundations." 

*f 

WILLIAM  M.  CAMP 

Editor  of  The  Railway  and  Engineering  Review;  American  Society  of  Civil  Engineers. 
Author  of  "  Notes  on  Track  Construction  and  Maintenance." 
^ 

W.  M.  PATTON 

Late  Professor  of  Engineering  at  the  Virginia  Military  Institute. 
Author  of  "A  Treatise  on  Civil  Ensrineerinsr." 


CHARTING  DIRECTLY  FROM  NATURE 

From  observations  just  made  through  alidade  trained  on  telemeter  rod  at  one  of  the  rod- 
men's  stations,  the  distance  and  bearing  of  that  station  are  determined.  Reduced  to  scale, 
this  data  corresponds  to  a  line  of  definite  length  and  direction,  which  the  topographer  is  here 
shown  drawing  directly  on  the  plane-table  sheet. 


Foreword 


HE  marvelous  developments  of  the  present  day  in  the  field 
of  Civil  Engineering,  as  seen  in  the  extension  of  railroad 
lines,  the  improvement  of  highways  and  waterways,  the 
increasing  application  of  steel  and  reinforced  concrete 
to  construction  work,  the  development  of  water  power 
and  irrigation  projects,  etc.,  have  created  a  distinct  necessity 
for  an  authoritative  work  of  general  reference  embodying  the 
results  and  methods  of  the  latest  engineering  achievement. 
The  Cyclopedia  of  Civil  Engineering  is  designed  to  fill  this 
acknowledged  need. 

C.  The  aim  of  the  publishers  has  been  to  create  a  work  which, 
while  adequate  to  meet  all  demands  of  the  technically  trained 
expert,  will  appeal  equally  to  the  self-taught  practical  man, 
who,  as  a  result  of  the  unavoidable  conditions  of  his  environ- 
ment, may  be  denied  the  advantages  of  training  at  a  resident 
technical  school.  The  Cyclopedia  covers  not  only  the  funda- 
mentals that  underlie  all  civil  engineering,  but  their  application 
to  all  types  of  engineering  problems;  and,  by  placing  the  reader 
in  direct  contact  with  the  experience  of  teachers  fresh  from 
practical  work,  furnishes  him  that  adjustment  to  advanced 
modern  needs  and  conditions  which  is  a  necessity  even  to  the 
technical  graduate. 


C.  The  Cyclopedia  of  Civil  Engineering  is  a  compilation  of 
representative  Instruction  Books  of  the  American  School  of  Cor- 
respondence, and  is  based  upon  the  method  which  this  school 
has  developed  and  effectively  used  for  many  years  in  teaching 
the  principles  and  practice  of  engineering  in  its  different 
branches.  The  success  attained  by  this  institution  as  a  factor 
in  the  machinery  of  modern  technical  education  is  in  itself  the 
best  possible  guarantee  for  the  present  work. 

C.  Therefore,  while  these  volumes  are  a  marked  innovation  in 
technical  literature  —  representing,  as  they  do,  the  best  ideas  and 
methods  of  a  large  number  of  different  authors,  each  an  ac- 
knowledged authority  in  his  work  —  they  are  by  no  means  an 
experiment,  but  are  in  fact  based  on  what  long  experience  has 
demonstrated  to  be  the  best  method  yet  devised  for  the  educa- 
tion of  the  busy  workingman.  They  have  been  prepared  only 
after  the  most  careful  study  of  modern  needs  as  developed 
under  conditions  of  actual  practice  at  engineering  headquarters 
and  in  the  field. 

C.  Grateful  acknowledgment  is  due  the  corps  of  authors  and 
collaborators  —  engineers  of  wide  practical  experience,  and 
teachers  of  well-recognized  ability  —  without  whose  co-opera- 
tion this  work  would  have  been  impossible. 


Table   of  Contents 


VOLUME  I 
PLANE  SURVEYING        .       ...       By  Alfred  E.  Phillips^        Page  *11 

Measurement  of  Lines  and  Areas  —  Plane  and  Geodetic  Surveying  —  Elementary 
Problems  —  Gunter's  Chain  —  Engineer's  Chain  —  Tape  —  Chaining  on  Slopes  — 
Correction  of  Errors  — Line  Running  —  Offsets  and  Tie-Lines  —  Field  Notes  — 
Vernier  (Direct,  Retrograde)  —  Leveling  Instruments  and  Leveling  —  Level 
Bubble  —  Hand-Levels  and  Clinometers  —  Leveling  Rods  —  Cross-Section  Rod  — 
Ranging  Poles  —  Y-Level  —  Line  of  Collimation  —  Instrumental  Parallax  — 
Spherical  and  Chromatic  Aberration  —  Adjustment  of  Instruments  —  Precise 
Spirit  Level  —  Setting  Up  the  Level  —  Care  of  Instruments  —  Problems  in  Level- 
ing —  Cross-Sectioning  —  Slope  Stakes  —  Land  and  Topographical  Surveying  — 
Meridian  Plane  —  Magnetic  Declination  —  Compass  and  Its  Adjustment  —  Tan- 
gent Scale  —  Angle-Measuring  Instruments  —  Bearing  —  Course  —  Relocation  of 
Lines  —  Farm  Surveying  —  Method  of  Progression,  of  Radiation,  of  Intersections 

—  To  Change  Bearing  —  Latitudes  and  Departures  —  Testing  and   Balancing  a 
Survey  —  Calculating  Contents  of  Surveyed  Areas  —  Double  Longitudes  —  Azi- 
muth —  Resurveys  —  True  Meridian  —  Surveyor's  Transit  —  Engineer's  Transit 

—  Tachymeter  —  Theodolite  —  Transit-Theodolite  —  Adjusting  and  Setting   Up 
Transit  —  Problems  in  Use  of  Transit  —  Traversing  —  Meander  Line  —  Stadia  — 
Stadia  Rods  —  Tables  of  Elongation,  Culmination,  and  Azimuth  of  Polaris  —  Base 
Map  of  the  United  States 

MECHANICAL  DRAWING  By  Ervin  Kenison       Page  211 

Instruments  and  Materials  —  T-Square  —  Triangles  —  Compasses  —  Dividers  — 
Bow- Pen  and  Pencil  —  Scales  —  Protractors  —  Irregular  Curves  —  Lettering  — 
Penciling  and  Inking  Plates  —  Geometrical  Definitions  —  Angles  <—  Surfaces  — 
Triangles  —  Quadrilaterals  —  Polygons  —  Circles  —  Measurement  of  Angles  — 
Solids  —  Pyramids  —  Cylinders  —  Cones  —  Spheres  —  Conic  Sections  —  Ellipse  — 
Parabola  —  Hyperbola  —  Odontoidal  Curves  —  Geometrical  Problems  —  Ortho- 
graphic Projection  —  Profile  Plane  —  Shade  Lines  —  Intersections  and  Develop- 
ments —  Isometric  Projection  —  Oblique  Projection  —  Line  Shading  —  Tracing 
-  Blue-Printing  —  Assembly  Drawing 

REVIEW  QUESTIONS Page  377 

INDEX Page  387 


*For  page  numbers,  see  foot  of  pages. 

tFor  professional  standing  of  authors,  see  list  of  Authors  and  Collaborators  at 
front  of  volume. 


PLANE  SURVEYING, 

PART  I. 


Surveying  is  the  art  of  determining,  from  measurements  made 
upon  the  ground,  the  relative  positions  of  points  or  lines  upon  the 
surface  of  the  earth  and  of  keeping  records  of  such  measurements  in 
a  clear  and  intelligent  manner  so  that  a  picture  (called  plat)  may  be 
made  of  the  lines  or  areas  included  in  the  survey.  The  records 
should  be  systematically  arranged  so  that  any  person  with  a 
knowledge  of  surveying  can  use  the  notes  intelligently.  The  field 
operations  consist  essentially  of  locating  points,  measuring  lines 
and  angles,  measuring  areas  and  laying  out  and  dividing  up  areas. 
It  is  apparent  that  Arithmetic  and  Geometry  are  essential  to  the 
successful  application  of  the  principles  of  Surveying. 

The  subject  may  be  divided  into  two  parts :  Plane  Surveying 
and  Geodetic  Surveying. 

In  Plane  Surveying,  the  portion  of  the  earth  included  in  the 
survey  is  regarded  as  a  horizontal  plane;  in  other  words,  the  curva- 
ture of  the  earth's  surface  is  neglected.  In  the  ordinary  operations 
of  land  surveying  this  assumption  will  not  cause  appreciable  error 
as  the  lines  and  areas  dealt  with  are  of  a  limited  extent. 

As  Geodetic  Surveying,  on  the  other  hand,  deals  with  extensive 
lines  and  vast  areas,  the  effect  of  the  curvature  of  the  earth's  sur- 
face must  be  taken  into  consideration. 

All  of  the  operations  of  surveying  must  proceed  from  the 
direct  to  the  indirect.  That  is  to  say,  we  must  first  measure 
directly  certain  quantities  upon  the  ground  and  from  these  calcu- 
late certain  other  quantities  that  cannot  be  measured  directly.  It 
is,  therefore,  apparent  that  all  field  measurements  must  be  made 
with  the  utmost  care,  consistent  with  the  nature  of  the  problem 
involved,  and  that  habitual  inaccuracy  and  slovenly  methods  of 
keeping  field  notes  must  be  avoided.  Full  details  accurately 
measured  and  carefully  and  systematically  recorded  should  be  the 
aim  of  every  engineer  who  would  ultimately  achieve  success. 

Copyright,  1908,  by  American  School  of  Correspondence. 


11 


PLANE  SURVEYING 


Measurement  of  lines.  Probably  the  most  elementary  prob- 
lem that  presents  itself  is  to  measure  the  horizontal  distance  be- 
tween two  points  without  the  use  of  instruments. 

This  can  best  be  done  by  pacing,  provided  both  points  are 
accessible.  In  order  to  make  this  method  of  measurement  efficient, 
it  is  necessary  to  determine  as  accurately  as  possible  the  length  of 
one's  pace.  To  do  this,  lay  off  upon  firm,  level  ground  by  any 
convenient  method,  a  line  from  50  to  100  feet  in  length.  Pass 
over  this  line  from  end  to  end,  back  and  forth,  keeping  careful 
account  of  the  number  of  steps  taken  each  time  the  distance  is 
covered.  The  total  distance  traversed,  divided  by  the  total  number 
of  steps  will  give  the  average  length  of  one's  pace.  In  thus 
ascertaining  the  length  of  the  pace  do  not  attempt  to  cover  three 
feet  at  every  step.  It  is  better  to  adopt  a  natural,  swinging  gait. 

Having  thus  determined  the  length  of  one's  pace,  the  distance 
between  two  points  may  be  measured  approximately  by  walking 
in  a  straight  line  from  point  to  point 
and  counting  the  number  of  steps. 

4A^--— Tg^ I     This  number  multiplied  by  the  length 
of  step  will  give  the  length  of  line 
required.     If  the  intervening  space 
*"    between  the  points  cannot  be  trav- 
ersed, as  for  instance  when  the  two 

points  are  on  opposite  sides  of  a  stream,  the  width  of  the  stream  may 
be  ascertained  -approximately  by  stationing  an  observer  on  each 
side  and  noting  the  time  elapsing  between  the  flash  of  a  pistol 
and  the  sound  of  the  report.  This  interval,  in  seconds,  multiplied 
by  1,090  (velocity  of  sound  in  feet  per  second)  will  give  the  dis- 
tance in  feet.  Proper  allowance  must  be  made  for  direction  and 
intensity  of  wind  and  therefore  measurements  of  this  kind  had 
best  be  made  upon  a  quiet  day. 

Another  elementary  problem  frequently  met  with  is  as  follows: 
Required  to  determine  the  altitude  of  an  object  such  as  a  house  or 
a  tree,  without  the  use  of  an  instrument. 

To  solve  this  problem,  take  an  ordinary  lead  pencil  and  hold 
it  in  a  vertical  position  about  two  feet  from  the  eye,  the  observer 
being  far  enough  from  the  object  for  the  visual  angle  intercepted 
by  the  pencil  to  just  cover  the  object  from  top  to  bottom.  The 


PLANE  SURVEYING 


observer  then  paces  the  distance  from  his  position  to  the  object. 
The  height  of  the  object  is  determined  as  follows  : 

Let  A,  Fig.  1,  represent  the  position  of  the  observer's  eye; 
BC  the  pencil  held  at  the  distance  AD  from  the  eye;  EF  the  object 
whose  height  is  to  be  ascertained.  AH  is  the  distance  from  the 
observer  to  the  object  and  is  to  be  paced.  Then  from  similar 
triangles  we  have 

BC  :  EF  :  :  AD  :  AH,  or  EF  -  B°  X  AH 

A.D 

For  example,  suppose  the  pencil  is  seven  inches  long  and  is 
held  at  a  distance  of  two  feet  from  the  eye;  the  distance  from  the 
observer  to  the  object  being  85  feet.  Then  from  the  formula 

7 
EF  = g —  =  24.8  feet  nearly. 

In  this,  as  in  other  problems,  all  quantities  should  be  reduced  to 
the  same  units. 

The  examples  just  given  must  be  understood  as  illustrations 
merely  and  the  student  should  avoid  slipshod  methods  ;  under- 
standing that  his  best  efforts  will  be  needed  in  all  surveying  prob- 
lems, and  that  the  best  is  none  too  good. 

SURVEYING  WITH  INSTRUMENTS. 

Gunter's  Chain,  so  called  from  the  inventor,  is  well  adapttd 
to  all  classes  of  problems  involving  the  calculation  of  areas  from 
lines  measured  in  the  field.  For  many  years  this  chain  has  been 
the  English  linear  unit  for  all  land  measurements.  It  should  be 
made  of  steel;  it  is  66  feet  or  4  rods  in  length  and  has  100  links, 
each  7.92  inches.  The  handles  are  fitted  with  swivels  to  prevent 
the  chain  from  kinking,  and  at  every  tenth  link  from  either  end  is 
attached  a  brass  tag  with  1,  2,  3  or  4  prongs  to  assist  in  measuring. 
Thus  the  tag  of  four  prongs  indicates  40  links  from  one  end,  (See 
Fig.  2)  but  it  represents  60  links  from  the  other  end;  therefore 
care  must  be  exercised  in  measuring,  or  distances  may  be  measured 
from  the  wrong  end  of  the  chain.  The  50-link  mark  is  round 
in  form  so  that  it  may  be  easily  distinguished  from  the  other  tags. 
Since  the  parts  are  called  links,  the  length  is  expressed  in  chains 


13 


PLANE  SURVEYING 


and  links;  it  is  written  thus:    15  chains  and  82  links  is    15.82 
chains. 

It  is  true  that  this  chain  is  rapidly  going  out  of  use,  yet  one 
should  be  thoroughly  acquainted  with  it,  because  many  of  the  land 
records  in  this  country  are  based  upon  it.  In  computing  areas, 
the  chain  has  the  advantage  that  square  chains  are  easily  reduced 
to  acres  by  simply  moving  the  decimal  point  one  place  to  the 


Fig.  2. 

left;  for  example,  a  chain  is  66  feet;  the  square  would  be  66x66— 
4356  square  feet,  which  is  TL  of  an  acre.  A  rectangular  lot  hav- 
ing two  sides  of  6.32  and  2.15  chains  respectively  =  13.5880 
square  chains  or  1.3588  acres. 

GUNTER'S  OR  LAND  MEASURE. 

7.92  inches 1  link 

100  links  or  66  feet  or  4  rods 1  chain 

10  square  chains  or  4  roods 1  acre  =  43560  square  feet 

640  acres 1  square  mile 

A  two-rod  or  half  chain  is  sometimes  used  instead  of  the  full 
Gunter's  chain.  Its  only  advantage  is  in  the  convenience  of  hand- 
ling a  shorter  chain  when  working  over  uneven  ground.  Formerly 
the  engineer's  chain  was  almost  universally  employed  in  making 
surveys  for  surface  canals,  sewers,  water- works  systems,  etc.  It 
differs  from  the  Gunter's  chain  in  that  is  100  feet  in  length  and  con- 
tains  100  links,  each  of  which  is,  therefore,  1  foot  long. 

The  unit  of  linear  measure  in   the  United  States  is  the  foot. 


14 


PLANE  SURVEYING 


In  measuring  lines,  a  chain  100  feet  long,  divided  into  100  links, 
is  now  in  use.  Distances  are  recorded  in  feet;  decimals  of  a  foot 
being  used  when  possible.  In  cities  where  accurate  and  precise 
measurements  are  necessary,  various  kinds  of  tapes  are  used  having 
the  foot  divided  decimally. 

It  has  been  decided  both  by  custom  and  law  that  the  length  of 
the  boundary  lines  of  a  field  is  not  the  actual  distance  on  the  surface 
of  the  ground,  but  is  the  projection  of  that  distance  on  a  horizontal 
plane.  The  area  of  a  field  is  not  the  exposed  superficial  surface, 
but  as  above  stated,  the  projection  of  that  surface  on  a  horizontal 
plane.  For  this  reason,  in  all  land  surveying,  horizontal  dis- 
tances are  to  be  measured  and  from  these  the  areas  computed. 

The  Gunter's  chain,  as  well  as  the  engineer's  chain,  is  a  very 
inaccurate  device  for  measuring  distances  and  areas  unless  special 
precautions  are  taken  to  counteract  the  errors  to  which  it  is  liable. 
Some  of  these  errors  are  cumulative  and  some  compensating,  and  in 
what  follows  no  attempt  will  be  made  to  classify  them.  Some  of 
the  causes  of  errors  will  be  pointed  out  and  the  surveyor  should 
do  all  in  his  power  to  eliminate  them. 

The  chain  will  sag  between  supports  and  thus  the  distance 
measured  will  be  too  short.  This  is  sometimes  allowed  for  by 
making  the  chain  a  given  amount  longer  than  the  standard.  Again, 
the  chain  may  be  standard  under  a  certain  pull  and  temperature, 
arid  for  very  precise  work  a  spring  balance  is  attached  to  one  end 
of  the  chain  to  register  the  pull.  A  thermometer  also  is  provided 
but  is  of  little  value  from  the  fact  that  the  temperature  of  the 
chain  may  vary  considerably  from  that  of  the  atmosphere.  Still 
further,  the  length  of  the  chain  is  likely  to  be  increased  from  the 
wear  of  the  links  and  connections.  Each  link  with  its  connec- 
tion has  six  wearing  surfaces  so  that  if  each  surface  is  worn  away 
but  y^j-  inch  the  chain  will  elongate  6  inches.  The  rings  and 
loops  at  the  end  of  the  links  are  frequently  stretched  out  of  the 
true  form,  thus  elongating  the  chain;  or  the  links  may  become 
bent,  thus  shortening  the  chain.  In  pulling  the  chain  over  the 
ground  the  links  and  rings  have  a  tendency  to  collect  weeds,  mud, 
etc.,  and  thus  shorten  the  chain.  In  cold  weather,  ice  and  snow 
may  collect  in  the  joints  with  the  same  result.  In  using  the  chain, 
the  links  and  rings  kink  and  twist,  and  a  sudden  jerk  may  break 


15 


8  PLANE  SURVEYING 

the  chain.  For  these  reasons  the  chain  is  not  at  present  used  as 
much  as  formerly. 

The  Tape.  Tapes  are  made  of  various  materials  and  are  known 
as  linen,  metallic  and  steel. 

Linen  tapes,  from  the  nature  of  the  material,  are  likely  to  twist 
and  tangle  and  when  wet  are  easily  stretched;  for  these  reasons 
they  do  not  long  retain  their  standard  length.  They  are  used  only 
in  the  roughest  kind  of  work.  Metallic  tapes  have  a  linen  body 
with  threads  of  copper  or  brass  running  throughout  their  length. 
These  metallic  threads  prevent  twisting  and  tangling  and  in  a  gen- 
eral way  assist  in  preserving  the  standard  length  of  the  tape.  They 
are  better  than  linen  tapes  but  not  suitable  for  "good"  work. 

Steel  tapes  are  of  two  kinds,  "ribbon"  and  "band."  Ribbon 
tapes  are  made  of  thin  steel  about  |  inch  wide.  They  are  usually 
made  in  lengths  of  50  or  100  feet.  They  are  divided  into  feet, 
tenths  and  hundredths  of  a  foot,  the  divisions  being  etched  upon 
the  tape.  The  other  side  of  the  tape  is  sometimes  divided  into  rods 
and  links  to  adapt  it  to  land  surveying,  and  it  is  either  wound  up 
into  a  leather  case  or  upon  a  reel. 

Ribbon  tapes  are  generally  used  when  considerable  accuracy 
in  measurements  is  required,  such  as  laying  out  foundations  for 
buildings,  bridge  piers,  measuring  up  sewer  lines,  etc.  From  the 
nature  of  their  construction,  they  will  not  stand  much  wear  and 
tear,  and  are  therefore  not  adapted  to  the  rough  usuage  of  general 
field  work.  If  carried  in  the  case  or  reel,  on  account  of  the  sharp 
bend  at  the  center,  the  tape  will  soon  break  off  at  that  point. 
After  use  in  the  field,  the  tape  should  be  carefully  wiped  off  and 
oiled  if  necessary,  as  the  rust  will  obliterate  the  graduations  and 
make  it  difficult  to  read.  In  using  the  ribbon  tape  in  the  field, 
care  must  be  exercised  to  prevent  twisting  and  kinking  or  catching 
under  sticks  or  stones,  as  a  slight  jerk  will  break  it. 

The  band  tape  is  best  adapted  to  general  field  work  and  to 
rough  usage.  It  is  made  of  heavy  steel  about  T56  of  an  inch  wide 
and  100  feet  long,  divided  into  feet;  usually  the  first  and  last  foot 
are  divided  into  tenths.  The  one-foot  divisions  may  be  marked 
by  rivets,  although  the  rivets  tend  to  weaken  the  tape.  They  are 
sometimes  marked  by  solder,  which  is  notched  at  the  proper  point 
and  stamped  with  the  number.  They  are  usually  fitted  with  light, 


PLANE  SURVEYING  9 

detachable  handles  for  use  in  the  field,  but  these  are  easily  displaced 
or  often  lost  in  dragging  the  tape  over  stones  or  through  grass.  It 
is  better  to  fit  the  tape  with  leather  handles  large  enough  to  easily 
go  over  the  hand.  After  use,  the  band  tape  should  be  gathered 
up  in  loops  about  three  feet  long  and  tied  in  the  middle  forming  a 
figure  eight.  If  it  is  desirable  to  wind  the  tape  upon  a  reel,  there 
are  at  present  upon  the  market,  several  styles  of  reels,  stiff  in  con- 
struction and  convenient  to  carry. 

The  tape,  like  the  chain,  is  likely  to  change 
in  length  due  to  changes  of  temperatures,  and 
unless  the  proper  pull  is  applied  to  the  ends 
it  will  measure  short  of  the  standard.  Al- 
together it  is  more  accurate  than  the  chain  and 
of  late  years  has  largely  replaced  it  for  all 
kinds  of  field  work.  Indeed,  with  proper 
precautions,  it  has  been  found  possible  to 
obtain  nearly  as  accurate  results  as  with  the 
most  elaborate  apparatus  designed  for  meas- 
uring lines. 

Since  the  methods  of  using  the  chain  in 
the  field  are  the  same  as  for  using  the  tape  it 
will  be  sufficient  to  explain  the  methods  of 
using  the  latter.  The  Tape. 

In  connection  with  the  tape  there  should  be  provided  a  set  of 
eleven  marking  pins  from  15  to  18  inches  in  length.  To  each  pin 
should  be  attached  a  piece  of  red  flannel  to  prevent  its  being  over- 
looked in  the  grass.  There  should  also  be  provided  two  rods 
(called  flags),  from  6  to  8  feet  in  length  divided  into  foot  lengths 
and  painted  alternately  red  and  white.  These  rods  are  sometimes 
constructed  of  straight  white  pine,  but  |-inch  gas  pipe  fitted  with 
a  steel  shoe'is  better.  It  is  desirable  also,  to  provide  a  plumb-bob 
and  string,  and  a  hatchet. 

Use  of  the  Tape  or  Chain.  For  measuring  a  line  with  the 
tape,  two  men  are  required,  a  "  leader "  and  "  follower,"  or  head 
and  rear  tapemen.  The  first  step  is  to  set  one  of  the  flags  at  the 
far  end  of  the  line  to  be  measured,  or  if  the  line  is  too  long,  only 
as  far  ahead  as  can  be  distinctly  seen.  It  is  best  to  mark  the 
beginning  of  a  line  with  a  stake  driven  as  closely  to  the  ground  as 


17 


10  PLANE  SURVEYING 

circumstances  will  permit.  The  tape  is  then  unrolled  or  unfolded 
in  the  direction  of  the  line,  the  100-foot  mark  going  ahead.  The 
leader  takes  the  pins  and  the  forward  end  of  the  tape  and  with  a 
flag  walks  off  in  the  direction  of  the  forward  end  of  the  line, 
dragging  the  tape  after  him.  When  nearly  one  hundred  feet  away? 
the  follower  cries  "  down  "  and  the  leader  faces  the  follower  holding 
the  flag  vertically  to  be  signalled  into  line  by  the  follower.  The 
tape  is  then  stretched  and  straightened  and  a  pin  stuck  vertically 
into  the  ground  exactly  at  the  100-foot  mark.  The  leader  then 
picks  up  his  end  of  the  tape  and  starts  off  as  before,  the  process 
being  repeated  each  time,  except  that  the  follower  must  be  particular 
to  pick  up  each  pin  that  is  left  in  the  ground  by  the  leader. 

If  the  line  is  more  than  eleven  tapes  in  length,  after  the 
leader  has  stuck  his  last  pin  he  cries  "  pins  "  and  the  follower 
delivers  to  him  the  ten  pins  that  he  has  picked  up.  If  the  line  to 
be  measured  is  very  long,  some  method  should  be  adopted  for 
keeping  count  of  the  number  of  times  the  pins  have  been  exchanged. 
If  the  line  ends  with  less  than  the  length  of  a  tape,  the  leader  pulls 
out  the  tape  to  its  full  length,  not  sticking  a  pin,  however,  and 
then  walks  back  and  notes  the  distance  from  the  last  pin  to  the  end 
of  the  line.  This  distance  added  to  the  number  of  pins  held  by 
the  follower,  including  the  last  one  stuck,  will  give  the  distance 
from  the  point  at  which  the  pins  were  exchanged.  For  instance, 
if  the  follower  has  six  pins  and  the  end  of  the  line  is  65  feet  from 
the  last  pin,  the  entire  distance  from  the  point  of  exchange  of  pins 
is  665  feet.  It  must  be  remembered  that  each  exchange  of  pins 
counts  for  10  tape  lengths  or  1,000  feet. 

Chaining  on  Slopes.  One  of  the  most  important  uses  of  the 
chain  is  to  measure  accurately  distances  where  the  surface  of  the 
land  is  uneven  or  of  a  sloping  nature.  In  measuring  up  or  down 
a  slope,  one  end  of  the  tape  is  raised  until  the  top  is  as  nearly  as 
possible  in  a  horizontal  plane.  If  the  slope  is  too  steep  to  permit 
of  one  end  of  a  full  tape  being  raised  enough  to  bring  the  tape 
horizontal,  the  tape  is  "broken,"  that  is  to  say,  only  a  part  of  the 
tape  is  used  at  each  measurement.  To  do  this  the  tape  should  be 
stretched  to  its  full  length,  the  leader  returning  to  such  a  point 
upon  the  tape  that  the  portion  between  himself  and  the  follower 
may  be  properly  leveled.  A  measurement  .is  made  with  this  por- 


18 


PLANE  SUKVEYING  11 

tion,  the  operation  being  repeated  with  the  next  section  of  tape  and 
so  on  until  the  entire  tape  has  been  used.  Care  should  be  taken 
not  to  confuse  the  pins.  The  high  end  of  the  tape  may  be  trans- 
ferred in  any  one  of  several  ways,  depending  upon  the  degree  of 
accuracy  required.  For  great  accuracy,  a  plumb-bob  should  be 
used  but  it  should  not  be  dropped  and  the  pin  placed  in  the  hole 
made.  It  should  be  placed  about  where  the  bob  will  drop  and  the 
grass  should  be  tramped  down  and  the  ground  smoothed.  The 
bob  should  then  be  lowered  carefully  until  it  almost  touches  the 
ground  and  allowed  to  come  to  rest.  Then  lower  it  until  it  reaches 
the  ground  when  the  pin  should  be  stuck  in  the  ground  slantwise 
across  the  line  exactly  at  the  point  of  the  bob.  If  less  accuracy  is 
permissible,  it  may  be  sufficient  to  drop  the  pin,  ring  end  down 
and  note  where  it  strikes  the  ground,  or  a  pebble  may  be  dropped 
in  the  same  way.  In  measuring  uphill,  the  follower  must  hold  the 
bob  directly  over  the  pin  in  the  ground  while  he  aligns  the 
leader  and  sees  that  he  sticks  the  pin  while  the  bob  is  directly 
over  the  point  in  the  ground.  It  is  much  easier  to  measure  down 
than  up  hill  so  that  when  close  measurements  are  required  on 
slopes,  the  measurement  should,  if  possible,  be  made  down  hill. 
Even  under  the  most  favorable  conditions  measuring  lines  with  a 
tape  is  a  most  difficult  operation  for  experts,  and  beginners  cannot 
expect  to  attain  efficiency  except  by  constant  practice  and  careful 
attention  to  every  detail  that  will  tend  to  eliminate  error.  For  the 
method  of  chaining  up  and  down  hill,  see  Fig.  3. 

Let  it  be  required  to  find  the  distance  A  M,  at  which  points 
two  hubs  have  been  established.  Let  A  B  C  D  E  F,  etc.,  be  the 
points  of  the  successive  chaining  and  a~bc  d,  etc.,  the  horizontal 
planes.  Starting  from  the  point  A;  suppose  the  surface  between 
A  and  B  to  be  of  no  great  difference  in  elevation,  therefore,  the  full 
length  of  the  chain  can  be  used  between  these  two  points.  The 
head  chainman  goes  to  the  point  B,  and  holds  the  head  end  of  the 
chain  on  the  ground,  while  the  rear  chainman  holds  the  zero  end 
at  A  and  with  aid  of  the  plumb-bob  "plumbs  down,"  thus  chain- 
ing the  distance  horizontal.  This  distance  measured,  the  head 
chainman  establishes  a  peg  in  the  ground  and  calls  out  the  dis- 
tance "One  hundred"  or  station  one,  then  goes  to  C,  which  must 
be  approximately  low  enough  to  allow  the  rear  chainman  to  con- 


19 


PLANE  SURVEYING 


veniently  plumb  down  to  B.  In  this  case  the  slope  of  the  hill  is 
greater  than  that  between  A  and  B,  thus  the  impracticability  of 
using  the  full  length  of  the  chain  is  apparent.  This  distance, 
therefore,  is  taken  at  a  fractional  part  of  the  chain,  as  before  stated, 
called  "breaking  the  chain."  The  distances  at  such  breaks  should 
always  be  taken  at  an  even  number  of  feet  whenever  possible  and 
at  distances  that  are  easily  remembered,  as  10,  20,  25,  50  feet,  etc. 
The  leader  in  every  instance  calls  out  the  distance  of  such  ubreaks" 
and  the  rear  chainman  goes  to  the  next  peg  and  holds  off  the  num- 
ber of  feet  previously  called  out.  Now  as  the  distance  A  B  is  100 
feet  and  the  distance  B  C  40  feet,  the  head  chainman  at  C  calls 
out  "1  plus  40,"  meaning  140  feet  from  A.  He  next  goes  to  D  and 
the  rear  chainman  calls  out  the  distance  measured  "1  plus  40"  and 
holds  off  40  feet  at  I,  and  plumbs  down  to  C.  In  this  case  the 
leader  also  plumbs  down  from  c  to  D.  This  method  is  continued 


until  M  is  reached,  using  the  system  of  1,  2,  3,  4,  etc.,  plus  the 
fractional  measured  distance,  instead  of  using  the  whole  number, 
as  125,  225,  etc.  The  rear  chainman  should  gather  the  pins  after 
a  new  point  has  been  established.  As  already  stated  the  chaining 
can  be  checked  by  counting  the  pins  picked  up.  Always  allow 
the  last  pin  to  remain  in  the  ground  until  absolutely  certain  it  is 
no  longer  desired  and  can  be  of  no  further  service.  It  is  im- 
portant that  the  distances  should  be  checked  by  both  chainmen  as  it 
may  prevent  serious  mistakes,  and  in  some  cases  prevent  rechain ing 
the  entire  distance.  Distances  in  places  where  angles  are  taken 
are  sometimes  checked  in  the  office  by  Trigonometry. 

A  few  hints  in  regard  to  the  use  of  the  tape  may  not  come 


Always  measure  to  and  from  the  same  side  of  a  pin. 
Hold  the  end  of  the  tape  as  near  the  ground  as  possible. 
Before  sticking  the  pin,  be  sure  there  are  no  kinks  in  the  tape 


20 


PLANE  SUKVEYING  13 

and  that   the   tape  is  not  deflected  to  one  side  by  grass,  sticks  or 
stones. 

Never  straighten  the  tape  with  a  jerk;  raise  it  clear  of  the 
ground  and  straighten  and  stretch  with  a  steady  pull  and  lower 
steadily  into  place. 

The  tape  man  should  never  brace  himself  against  a  pin;  he 
should  assume  a  position  of  stable  equilibrium,  preferably  with  one 
hand  upon  the  ground. 

In  passing  over  uneven  ground,  every  reasonable  effort  should 
be  made  to  hold  the  tape  level.  Too  much  time  should  not  be 
spent  in  attempting  to  hold  the  end  of  the  tape  exactly  over  the 
point  in  the  ground  when  the  difference  of  level  of  the  ends  of  the 
tape  is  sufficient  to  neutralize  what  would  otherwise  be  considered 
an  accurate  measurement. 

In  passing  over  rough  ground  the  tape  should  be  carried  free 
from  the  ground,  thus  saving  it  unnecessary  wear.  The  length  of 
the  tape  is  likely  to  vary  from  time  to  time,  from  changes  in  tem- 
perature, from  constant  stretching  and  from  accident  in  the  field. 
For  this  reason  the  surveyor  should  compare  frequently  the  lengths 
of  his  tapes  with  that  of  a  standard.  The  length  of  the  standard 
tape  may  sometimes  be  conveniently  laid  off  upon  the  floor  of  a 
building,  or  two  monuments  may  be  set  in  the  ground,  the  proper 
distance  between  them  being  measured  either  by  a  standardized 
tape  or  by  means  of  wooden  rods.  Having  found  the  error  in  the 
length  of  the  tape  the  necessary  corrections  can  then  be  made.  If 
a  line  has  been  measured  upon  the  ground,  and  it  is  afterwards 
found  that  the  length  of  the  tape  is  in  error,  the  true  length  of 
the  line  may  be  found  from  the  following  proportion:  the  true 
length  of  the  tape  is  to  the  length  of  the  standard  tape,  as.  the  true 
length  of  the  line  is  to  the  length  of  the  line  as  measured. 

Suppose  a  line  as  measured,  is  found  to  be  025  feet  in  length 
and  it  is  afterwrards  found  that  the  tape  is  too  long,  by  six  inches. 
Then  we  have:     100.5  :  100  ::  x  :  625 
from  which  x  =  true  length  of  line  =  628^  feet. 

EXAMPLES  FOR  PRACTICE. 

1.  A  line  as  measured  with  a  certain  tape  is  580  feet  in 
length.  It  is  afterward  found  that  the  tape  is  -^  of  a  foot  too 
short.  Determine  the  true  length  of  the  line.  Ans.  578.26  feet. 


14  PLANE  SURVEYING 


2.  A  line  is  known  to  be  840  feet  in  length,  but  when  meas- 
ured with  a  certain  tape  is  found  to  be  842£  feet  in  length.     Deter- 
mine the  true  length  of  the  tape. 

Ans.  99.7  feet. 

3.  A  certain  field  was  measured  with  a  Gunter's  chain  and 
found  to  contain  625  acres.     It  was  afterwards  found  that  the  chain 
was  ^  foot  too  Ions.     Determine  the  true  area  of  the  field. 

Ans.  629.74  acres. 

If  an  area  has  been  measured  with  a  certain  tape  that  is  after- 
wards  found  to  be  in  error,  thecorrected  area  may  be  found  by  the 
following  proportion :  The  square  of  the  true  length  of  the  tape  is 
to  the  square  of  the  length  of  the  standard  tape  as  the  true  area  is 
to  the  measured  area. 

Examples  will  now  be  given  illustrating  the  use  of  the  chain 
or  tape,  in  the  field. 

1.  To  erect  a  perpendicular  at  a  given  point  in  a  line. 
Let  AB  Fig.  4  be  the  given  line  and  C  the  point  in  the  line  at 
which  it  is  desired  to  erect  a  perpen- 
dicular. Since  a  triangle  formed  of 
the  sides  3,  4  and  5  (or  any  multiple  of 
these)  will  con  tain  a  right  triangle,  take 
parts  of  the  chain  or  tape  representing 
these  distances  or  multiples  and  have 
the  angle  included  between  the  shorter 
sides  at  C.  Therefore,  fasten  one  end 
of  the  tape  or  chain  at  E,  30  links  or 
feet  from  C  and  the  90th  link  or  foot 

at  C.  Then  with  the  50-foot  mark  in  one  hand,  wralk  away  from  BC 
until  both  of  the  segments  DE  and  DC  are  taut.  Stick  a  pin  or 
stake  at  D  and  DC  will  be  the  perpendicular  required.  If  the 
perpendicular  should  be  longer  than  can  be  laid  out  with  the  tape 
or  chain,  lay  out  CD  as  described  and  align  a  "flag"  from  C  to  D 
produced. 

2.  To  let  fall  a  perpendicular  to  a  given  line  from  a  given 
point  outside  the  line,  (a)  When  the  point  is  accessible: 

Let  AB  Fig.  5  be  a  given  line  and  C  a  point.  From  C  as  a 
center,  with  any  convenient  length  of  tape  or  chain  as  a  radius, 
describe  the  arc  DE,  cutting  the  given  line  at  points  D  and  E. 
Stick  pins  at  D  and  E  and  measure  the  distance.  Bisect  this  dis- 


PLANE  SURVEYING 


15 


tance  at  F;  then  OF  will  be  the  perpendicular  required.  If  the 
line  AB  is  too  far  from  0  to  be  reached  with  the  chain  or  tape,  it 
will  be  necessary  to  range  out  a  line  conveniently  near  to  C  which 
shall  be  parallel  to  AB.  To  do  this  erect  at  any  convenient  point 
on  AB,  as  at  N,  Fig.  6,  the  perpendicular,  and  prolong  it  as  far  as 
necessary,  as  R.  At  R,  erect  RS  perpendicular  to  RN.  Then  the 
perpendicular  let  fall  from  C  upon  RS  and  prolonged  to  AB  will 
be  perpendicular  to  AB. 


Fig.  5. 


Fig.  6. 


(b)  When  the  point  is  inaccessible:  Let  AB,  Fig.  7,  be  the 
given  line  and  C  the  inaccessible  point  from  which  it  is  desired  to 
drop  the  perpendicular  to  the 
line  AB.  At  any  convenient 
point  F  in  AB  erect  the  per- 
pendicular  FD  and  extend  FD 
to  E,  so  that  FE=FD.  Locate 
the  point  B  so  that  B,  D  and  C 
will  be  in  the  same  straight  line. 
Sight  from  E  to  C  and  find  the 
point  H  in  which  this  visual  line 
crosses  AB.  Next  find  the  point 
G  at  the  intersections  of  DH  and 
BE  prolonged.  Sight  from  G  to 
C  and  the  point  M  in  which  this 
visual  line  crosses  AB  will  be  the 
point  required  and  the  distance 
MG  will  equal  MC.  MC  will-be  Fig.  7. 

the  perpendicular  to  AB  at  M. 

3.  Through  a  given  point  to  run  a  line  that  shall  be 
parallel  to  the  given  line.  The  given  point  and  given  line  being 
accessible:  Let  C,  Fig.  8,  be  the  given  point  and  AB  the  given  line. 


16 


PLANE  SUEVEYING 


From  point  C  let  fall  CD  perpendicular  to  AB.     At  C  erect  OF 
perpendicular  to  CD;  then  EF  will  be  the  parallel  required. 

4.  To  prolong  a  line  beyond  an  obstacle.  Let  AB,  Fig.  9, 
be  the  given  line  which  is  intercepted  by  a  tree,  house  or  other  ob- 
stacle. It  is  required  to  locate  the  line  CD  which  will  be  in  the 
direction  of  AB  produced.  At  B  erect  BE  perpendicular  to  AB 
of  sufficient  length  to  clear  the  obstacle  and  at  E  erect  EF  perpen- 
dicular to  EB  prolonging  EF  beyond  the  obstacle.  At  F  and  C 
erect  perpendiculars  to  EF  and  CF  making  CF  equal  in  length  to 
BE,  then  CD  will  be  the  line  required  and  the  distance  from  A  to 
D  will  equal  AB  plus  EF  plus  CD. 


Fig.  8.  Fig.  9. 

5.  When  both  ends  of  a  line  are  accessible,  but  the  line 
cannot  be  measured  directly,  on  account  of  obstacles. 

At  each  end  of  the  line  erect  perpendiculars  of  equal  length 
sufficient  to  clear  the  obstacles,  and  measure  the  length  of  the  line 
between  the  extremities  of  these  perpendiculars. 

6.  When  both  ends  of  a  line  are  accessible,  but  neither  can 
be  seen  from  the  other,  thus  preventing  direct  alignment. 

Such  a  case  occurs  when  it  is  desired  to  run  a  line  across  a 
wooded  field,  the  trees  and  un- 
derbrush preventing  the  align  - 
ment  of  the  intermediate  sta- 
tions. Let  AB  Fig.  10  be  the 
line  whose  length  is  desired. 


Fig.  10. 


From  A  run  a  line  AB'  (called 
a  random  line)  in  any  conven- 
ient direction  and  continue  it  till  the  point  B  can  be  seen  from  BY 
At  B'  erect  the  perpendicular  BB'  to  AB'  and  measure  BB'.  Then 
from  the  right-angled  triangle  ABB'  we  will  have 


24 


PLANE  SURVEYING 


17 


The  distance  from  A  to  any  intermediate  station  as  0  can  be 
found  by  measuring  the  length  of  the  perpendicular  CO'  to  AB' 
From  similar  triangles  we  have 

AC  :  CC'::AB  :  BB' 
CC'XAB 


or  AC  = 


BB' 


7.     To  locate  points  in  a  line  over  a  hill,  both  ends  of 
which  are  visible  from  points  near  the  summit. 


Fig.  11. 

Set  a  flag  at  each  of  the  points  A  and  B  Fig.  11.     One  man 
then  goes  to  D,  as  closely  in  line  with  A  and  B  as  can  be  esti- 
mated.    He  then  signals  a  man  at  C  in  line  with  A.     C  then 
signals  D  to  D'  in  line  with 
B.     D'  signals  C   to  C'  in 
line  with  A  and  so  on  alter- 
nately until  the  points  C"  and 
D"  are  reached  in  line  with 
A  and  B. 

If  the  points  A  and  B  can- 
not be  seen  from  the  top  of 
the  hill,  run  a  random  line 
over  the  hill  as  described  in 
problem  6  and  offset  to  the 
true  line. 

8.     To  locate  points  in  a  line  across  a  wide, 
the  extremities  of  the  line  being  visible  from  each  other. 

Fix  a  point  C,  Fig.  12,  on  the  edge  of  the  slope  in  line  with 
A  and  B.     Then  holding  a  plumb-line  at  C  and  sighting  across 


Fig.  12. 


25 


18 


PLANE  SUKVEYING 


Mg.  13. 


to  B  the  intermediate  points  D,  E,  F  and  C  can  be  put  into  line. 

1.  The  Field  Work  of  Measuring   Areas.      Let  us   con- 
sider the  triangular  field  ABC,  Fig.  13.      Beginning  at  any 
convenient  corner  as  A,  measure  from  A  to  B,  then  from  B  to  C, 
and  finally  from  C  to  the  point  of  beginning.     Should  a  stream 
cut  across  the  field  as  shown,  measurements  should  be  made  from 
the  corners  to  the  points  where  stream  crosses  the  boundary  lines. 

Should  it  be  found  impossible  to  measure  the 
sides  of  the  field  directly,  owing  to  zigzag 
fences  or  other  obstacles,  offset  parallel  lines 
as  in  the  figure  and  measure  the  length  be- 
tween such  parallels. 

The  area  of  the  figure  may  be  found  from 
the  following  rule:  From  one-half  the  sum  of 
the  three  sides,  subtract  each  side  separately. 
Multiply  together  the  half  sum  and  the  three 
remainders  and  extract  the  square  root  of  the 
product.  This  rule  is  explained  in  Art.  198  of  Elementary  Alge- 
bra and  Mensuration. 

If  the  lengths  of  the  sides  are  given  in  chains,  the  area  will 
be  given  in  square  chains.  If  the  lengths  of  the  sides  are  in  feet, 
the  result  will  be  in  square  feet. 

2.  To  survey  a  four-sided  field  with  the  tape  or  chain. 
Measure  around   the   field   in  the  same   way  as   before,    but  in 
addition,  it  will  be  necessary  to  measure  a  tie- 
line  between  two  opposite  corners,  thus  dividing 

the  figure  into  two  triangles,  the  sum  of  whose 
areas  will  give  the  area  of  the  entire  figure. 
Such  a  tie-line  is  shown  in  Fig.  14  by  the  dotted 
line  DB.  If  neither  of  the  diagonals  DB  nor 
AC  can  be  conveniently  measured,  measure  the 
short  tie-line  LS  for  instance,  and  the  distances 
CS  and  CL.  Then  in  the  triangle  LCS  the 
three  sides  are  given,  from  which  to  find  the 
angle  LCS.  Having  this  angle,  we  can  calcu- 
late the  length  of  DB  and  therefore  the  area  of  the  two  triangles 
composing  the  field. 

If  it  is  not  convenient  to  measure  the  tie-lines  inside  the  field, 


Pig.  14. 


PLANE  SUKVEYING  19 

two  adjacent  sides  as  BA  and  DA  can  be  prolonged  to  M  and  N 
forming  the  tie-line  MN.  It  will  usually  be  found  more  convenient 
to  lay  off  CS  equal  to  CL,  thus  forming  an  isosceles  triangle, 

3.  To  suwey  a  five-sided  field  with  the  tape  or  chain.  In 
this  case  two  diagonals  as  EB  and  BD,  Fig.  15,  or  two  tie-lines  as 
es  and  mn  must  be  measured  in  addition  to  the  lengths  of  the 

t> 

sides.  Whatever  the  number  of  sides,  a  sufficient  number  of 
diagonals  or  tie-lines  should  be  measured  to  divide  the  area  into 
triangles  from  which  the  area  of  the  entire  field  may  be  calculated, 
If  N  represents  the  number  of  sides  of  a  field,  there  will  be 
required  N-3  diagonals  or  tie- lines,  form- 
ing  N-2  triangles. 

To  simplify  calculations  when  tie-lines 
are  used  in  place  of  the  long  diagonals, 
the  following  method  may  be  adopted: 

Measure  off  Am  any  fractional  por- 
tion of  AE,  and  An  the  same  fractional 
portion  of  AB  and  measure  mn.  Then 
mn  will  be  to  EB  as  A;/?,  is  to  AE  or  as  An 
is  to  AB.  Suppose  for  example  that  Am 
is  TL  of  AE  and  An  is  J^-  of  AB.  There-  pjg  15 

fore  EB  is  10  times  the  length  of  mn. 

EXAMPLES  FOR  PRACTICE. 

1.  Given  the  three  sides  of  a  field  as  5.25,  6.50  and  4.60 
chains.     Find  the  area  of  the  field  in  acres,  and  square  rods. 

Ans.  1  acre,  31.52  square  rods. 

2.  Given  CB=3.65  chains,  CD=2.85  chains,  C*=C«?=0.50 
chains  and  t?«=0.65  chains.*     Calculate  the  area  of  the  triangle 
BCD.  Ans.  5.14  square  chains  area. 

Off=sets  and  Tie=Iines.  To  find  the  area  of  a  field  which  is 
bounded  in  part  by  a  stream,  it  is  necessary  to  use  off-sets,  as 
follows:  Measure  the  sides  of  the  field  in  the  usual  manner  and  for 
the  irregular  boundary  run  a  straight  line,  as  ED,  Fig.  16,  and 
calculate  the  area  of  the  field  included  between  these  boundary  lines. 
To  this  area  must  be  added  the  area  included  between  the  line  ED 
and  the  irregular  boundary. 

*NOTE;  In  all  problems  involving  the  measurement  of  land,  the  chain 
referred  to  is  the  Gunter's  chain  of  66  feet,  unless  otherwise  noted. 


27 


PLANE  SURVEYING 


To  find  this  area,  at  points  along  ED,  erect  perpendiculars  to 
the  irregular  shore  line  at  such  distances  that  the  lines  1'  2',  2'  3' 
etc.,  may  be  considered  straight.  The  desired  area  will  evidently 
equal  the  sum  of  the  areas  of  the  trapezoid?  thTi3  formed.  The 
distance  from  E  to  any  point  1,  2  or  3  on  ED  is  called  the  abscissa 
of  that  point  and  the  perpendicular  distances  from  ED  to  1'  2'  3' 
etc.,  are  called  the  ordi  nates  of  the  point. 


Pig.  16. 

Instead  of  summing  the  trapezoid  as  above,  the  desired  aroa 
may  be  found  by  the  following  rule:  Multiply  the  difference 
between  each  ordinate  and  the  second  succeeding  one  by  the 
abscissa  of  the  intervening  ordinate.  Multiply  also  the  sum  of  the 
last  two  ordinates  by  the  last  abscissa;  one-half  of  the  algebraic 
sum  of  these  several  products  will  be  the  area  required. 

To  lind  the  area  of  an  inaccessible  swamp,  a  lake  or  other 
area,  run  a  series  of  straight  lines  entirely  enclosing  the  given 
area,  and  since  the  diagonals  cannot  be  measured,  measure  tie-lines 
either  inside  or  outside  of  the  area.  As  already  stated,  calculate 
the  area  included  between  the  straight  boundary  lines  and  from 
this  area  substract  the  area  included  between  off-sets  let  fall  from 


PLANE  SURVEYING 


21 


points  upon  these  boundary  lines.  Reference  to  Fig.  17  will  make 
the  method  of  procedure  plain.  Surround  the  inaccessible  area 
by  straight  lines,  AB,  BC,  CD,  etc.,  and  calculate  the  enclosed 
area.  At  proper  intervals  along 
these  straight  lines,  erect  and 
measure  perpendiculars  ex- 
tending to  the  edge  of  the  inac- 
cessible area.  Compute  the  area 
between  these  perpendiculars 
by  the  rule  on  page  20  and  for 
the  required  area,  subtract  it 
from  the  area  previously  found. 
Since  the  long  diagonals  are  not 
accessible,  measure  the  area  by 
measuring  the  interior  tie-lines ; 
remembering  that  the  required 
number  of  tie-lines  will  be  less 
by  3  than  the  number  of  sides 
enclosing  the  area. 

Example.     Given    the    dis- 
tances  measured   along   the 

straight  line  AB(Eig.  18)  with  the  corresponding  off-sets  measured 
to  the  broken  line  ACDE.  It  is  required  to  compute  the  area 
between  A  B  and  the  broken  line  ACDE. 


Fig.  18. 


Difference  of  1st  and  3rd   ordinates  =  0'— 55'=— 55'  etc. 
"   2nd    "    4th  =40'— 35'=+  5' 

"   3rd    "    5th  =55'— 18'=+37' 

"   4th    "    6th  =35'— 40'=—  5' 

"   5th    "    7th  "         =18'— 60'=^2' 

Sum  of  last  two  ordinates  =  40/+60'=  lOCX 


PLANE  SURVEYING 


Abscissa  of  intermediate  ordinates  between  1st  and  3rd=  40*  X—  55'=-2200 

2nd  "  4th=  90'  X+  5'=  450 
3rd  "  5th=132'X+37'-=  4884 
4th  "  6th=172'X  —  5'=-  860 
5th  "  7th=217'X  —  42'=-9114 
«  last  ordinate  =  267'  XlOO'=26700 

One-half  the  algebraic  sum  of  the  products  as  given  above 
will  give  the  required  area. 

32034  —  12174 
Area  =  —  ~  —  =  9930  square  feet. 

EXAHPLE  FOR  PRACTICE. 

1.  Given  the  distances  measured  along  the  straight  line  AB 
Fig.  A  with  the  corresponding  off-sets  measured  to  the  broken  line 
ACDEFB,  to  find  the  area  between  AB  and  the  broken  line 
ACDEP^B.  Check  the  result  by  calculating  the  areas  of  the 
trapezoids  and  triangles  of  the  figure.  Ans.  11,875  square  feet. 


38- 


Keeping  the  Field  Notes.  In  keeping  field  notes,  clearness 
and  fullness  should  be  constantly  kept  in  mind.  As  field  notes 
often  pass  into  the  hands  of  a  second  party,  they  should  admit  of 
but  one  interpretation  to  a  person  at  all  acquainted  with  the  nature 
of  the  work.  Extra  time  spent  in  the  field  in  acquiring  data  will 
avoid  confusion  and  vexatious  delays  when  the  notes  are  worked 
up  in  the  office.  Avoid  the  habit  of  keeping  notes  upon  scraps  of 
paper  or  in  vest-pocket  note  books.  Provide  note  books  especially 
adapted  to  the  keeping  of  field  records  and  number  and  index  them 
so  that  the  contents  may  be  understood  at  a  glance.  Remember 
that  sketches  made  upon  the  ground  aid  materially  in  interpreting 
field  notes  that  otherwise  might  be  unintelligible. 

There  are  three  principal  methods  of  keeping  field  notes;  first, 
by  sketches  alone;  second,  by  notes  alone;  and  third  by  full  notes 


RODMAN  OF  A  PLANE-TABLE  PARTY  AT  A  STATION 

The  divisions  on  the  telemeter  rod.  read  between  the  cross-hairs  of  the  surveyor's  telescope, 
indicate  the  distance  of  the  rod  from  the  instrument. 


PLANE  SURVEYING 


23 


850 
750' 


supplemented  by  sketches.  The  third  method  is  without  doubt 
the  best,  but  examples  of  the  others  will  be  given.  For  keeping 
the  notes  of  the  chain  survey  there  should  be  provided  what  is 
known  as  a  field  book,  a  pencil  (preferably  411),  rubber  eraser  and 
and  a  short  rule  for  drawing  straight  lines. 

First*  by  Sketches  Alone.   Either  page  of  the  note  book  may  be 
used  for  sketching  but  it  will  be  more  convenient  to  use  the  right- 
hand  page,  as  it  is  ruled  into  squares,  thus  permitting  sketching  to 
scale.    Always  sketch  in 
the  direction  of  the  sur-  ,.. 

vey,  beginning  at  the 
bottom  of  the  page  and 
making  the  center  line 
of  the  page  correspond 
approximately  with  the 
North  and  South  lines. 

Second,  by  Notes 
Alone.  Use  the  left- 
hand  page  of  the  note 
book  beginning  at  the 
bottom  as  before.  Do 
not  crowd  the  notes,  and 
if  necessary  use  two  or 
more  pages.  See  Fig.  19. 

Fig.  19  shows  the 
method  of  keeping  the 
notes  of  the  survey 
shown  in  Fig.  20 


400' 
330' 

-B- 


500' 


470' 
420' 


695' 


1065' 
470' 
400' 


805' 
740' 


070' 
600' 


475' 
415' 


Fig.  19. 


Third,  by  Notes  and  Sketches.     It  is  apparent  that  in  this 
method  both  the  first  and  second  methods  are  embodied  in  the  notes. 


THE  VERNIER. 

The  vernier  is  an  auxiliary  scale  for  measuring  with  greater 
precision  the  spaces  into  which  the  principal  scale  is  divided.  The 
smallest  reading  of  the  vernier,  or  the  least  count,  is  the  difference 
in  length  between  one  division  on  the  main  scale  and  one  on  the 
vernier. 

A  vernier  is  said  to  be  direct  when  the  divisions  on  the 


31 


24 


PLANE  SURVEYING 


vernier  are  smaller  than  those  on  the  main  scale  Fig.  21  A;  retro- 
grade, when  the  divisons  on  the  vernier  are  greater  than  those  on 
the  main  scale.  See  Fig.  2 IB. 

In  Fig.  22  let  MM 
represent  a  scale  divided 
into  tenths;  then  since 
ten  spaces  on  the  vernier 
W  are  equal  to  nine 
spaces  upon  the  scale,  it 
is  evident  that  each  space 
upon  W  is  short  by  one- 
tenth  of  a  space  of  MM. 
The  least  count  is  there- 
fore, TV  of  TV  or  TJ0. 

The  vernier  and  slow 
motion  screw  of  the  ver- 
tical arc  of  the  engi- 
neer's transit  are  attach- 
ed to  the  left  hand 
standard  of  the  instru- 
ment. 

Fig.  23  represents  a 
vernier  as  applied  to  an 
engineer's  transit.  It 
will  be  noticed  that  the 
main  scale  is  divided  so 


A 

Fig.  20. 


as  to  read  directly  to  30  minutes.  The  vernier  is  so  divided  that 
29  spaces  upon  the  main  scale  equal  30  spaces  upon  the  vernier, 
therefore  the  least  count  of  the  vernier  is  -J^  of  30  minutes  or 
1  minute. 

A 


Mill 


1 1 1 1 1 1 1 1 1 1 1 


^ 


Fig.  21. 

It  will  be  apparent,  therefore,  that  the  readings  are  taken 
in  the  direction  of  the  increasing  graduations  of  the  main  scale. 
Thus,  for  example,  in  Fig.  23,  it  will  be  noted  that  the  zero  a. 


PLANE  SURVEYING 


25 


has  passed  the  156th  space  on  the  main  scale,  and  is  near  the  30 
minute  (half  degree)  division  &,  therefore  the  coinciding  lines  of 
the  vernier  and  main  scale  must  be  between  0  and  30',  and  we 

123456789 


M 


M 


234567      89     10 

Fig.  22. 


find  them,  by  looking  along  the  scale  of  the  vernier,  at  17  minutes 
hence,  the  reading  is  156°  00' +  17'=  156°  17'. 


Fig.  23. 

Fig.  24  represents  another  method  of  division  of  the  circle  of 
the  transit.  The  vernier  is  double,  and  the  figures  on  the  vernier 
are  inclined  in  the  same  direction  as  the  figures  on  the  scale  to 
which  they  belong. 

*,0  »0 


Fig.  24. 

It  will  be  noticed  that  the  main  scale  reads  directly  to  20 
minutes  and  that  the  vernier  is  so  divided  that  39  spaces  upon  the 
scale  correspond  to  40  spaces  upon  the  vernier.  The  least  count 
of  the  vernier  is  therefore  -^  of  20  minutes  or  |^  of  1  minute 
equals  30  seconds. 

To  read  the  inside  scale,  it  will  be  noticed   the  zero  of  the 


PLANE  SURVEYING 


vernier  is  beyond  the  138°  mark  and  about  half  way  between  the 
first  and  second  20'  divisions.  The  reading  so  far  is  then  138°'20'. 
Now  look  along  the  vernier  to  the  right  until  a  line  upon  the  ver- 
nier is  found  that  seems  to  be  a  prolongation  of  a  line  upon  the 
scale.  This  occurs  at  the  division  marked  10  upon  the  vernier 
so  that  the  reading  is  138°  20'  +  10'  or  138°  30'. 

For  the  outside  scale,  the  zero  of  the  vernier  is  beyond 
the  221°  mark  and  about  half  way  between  the  first  and  second 
20'  divisions.  The  reading  so  far  is  therefore  221°  20'.  Now 
look  along  the  vernier  to  the  left  as  before,  and  the  divisions  coin- 
cide at  the  division  marked  10  upon  the  vernier,  so  that  the  read- 
ing of  the  outside  scale  is  221°  20'  -f  10'  or  221°  30'.  The  sum 
of  the  readings  of  the  two  scales  equals  300°  as  it  should. 

EXAMPLES  FOR  PRACTICE. 

1.  Determine  the  least  count  of  the  vernier  in  Fig.  A,  39 
spaces  upon  the  scale,  being  equal  to  40  spaces  upon  the  vernier. 

2.  Determine  the  least  count  of  the  vernier  in   Fig.  B,  59 
spaces  upon  the  scale  being  equal   to  60  spaces  upon  the  vernier. 
The  figure  represents  what  is  called  2k  folding  vernier.     To  read  it 
follow  along  the  vernier  in  the  usual  way  until  the  division  marked 
10  is  reached.     If  there  are  no  corresponding  lines,  then  go  back 
to  the  other  end  of  the  vernier  beginning  with  the  other  10  mark 
and  follow  it  back  toward  the  center  of  the  vernier. 

3.  Determine  the  least  count  of  the  vernier  of  Fig.  C,  which 
represents  the  usual  method  of  dividing  the  vertical  circle  of  the 
transit. 

The  Level  Bubble  is  one  of  the  most  important  attachments  of 
an  engineering  instrument,  and  an  instrument  otherwise  good  may 
be  rendered  useless  by  imperfect  level  tubes. 

The  spirit  level  is  a  glass  tube  nearly  tilled  with  a  mixture  of 
ether  and  alcohol,  the  remaining  space  being  occupied  with  the 
vapor  of  ether.  Alcohol  alone  has  not  proved  satisfactory  as  it  is 
too  sluggish  in  its  movements,  thereby  rendering  an  instrument 
lacking  in  sensitiveness.  If  the  tube  were  perfectly  cylindrical, 
the  bubble  would  occupy  the  entire  length  of  the  tube,  when  hor- 
izontal, or  when  slightly  inclined  to  the  horizon,  thus  rendering  it 
impossible  to  tell  when  the  tube  is  in  a  truly  horizontal  position. 


34 


PLANE  SURVEYING 


The  tube  is,  therefore,  ground  on  the  inside  so  that  a  longitudinal 
section  is  a  segment  of  a  circle.  If  the  tube  is  not  ground  to  an 
an  even  curvature  the  bubble  will  not  travel  the  same  distance  for 
every  minute  of  arc  to  the  extreme  ends  of  the  tube,  and  an  other- 
wise perfect  instrument  will  not  work  well. 

**, 

B 


Fig.  C. 

A  line  tangent  to  the  circular  arc  at  its  highest  point,  as  indi- 
cated by  the  middle  of  the  bubble,  or  a  line  parallel  to  this  tangent, 
is  called  the  axis  of  the  bubble  tube.  This  axis  will  be  horizontal 
when  the  bubble  is  in  the  center  of  the  tube.  Should  the  axis  be 
slightly  inclined  to  the  horizontal,  the  bubble  will  move  toward 
the  higher  end  of  the  tube,  and  the  movement  of  the  bubble  should 
be  proportional  to  the  angle  made  by  the  axis  with  the  horizontal. 
Therefore  if  the  tube  is  graduated,  being  a  portion  of  the  circum- 
ference of  a  circle,  with  a  radius  so  large  that  the  arc  of  a  few  sec- 


35 


28  PLANE  SURVEYING 

onds  is  of  an  appreciable  length,  it  will  be  possible  to  determine 
the  angle  that  the  axis  may  make  at  any  time  with  the  horizontal, 
provided  the  angular  value  of  one  of  the  divisions  of  the  tube  is 
known.  This  is  done  by  noting  by  how  many  divisions  the  center 
of  the  bubble  has  moved  from  the  center  of  the  tube. 

Since  divisions  of  uniform  length  will  cover  arcs  of  less  angu- 
lar value  as  the  radius  of  the  tube  increases,  and  since  a  bubble 
with  a  given  bubble  space  will  become  more  elongated  as  the  radius 
is  increased,  the  sensitiveness  of  the  bubble  is  proportional  to  the 
radius  of  curvature  of  the  tube  and  the  length  of  the  bubble.  The 
length  of  the  bubble,  however,  will  change  with  changes  in  tem- 
perature, becoming  longer  in  cold  weather  and  shorter  in  warm 
weather.  This  is  due  to  the  fact  that  the  liquid  in  the  tube  expands 
and  contracts  more  rapidly  than  the  glass.  If  the  bubble  contracts 
excessively,  the  sensitiveness  is  thereby  impaired,  and  it  should 
be  possible  to  regulate  the  amount  of  liquid  in  the  tube.  This  is 
done  by  means  of  a  partition  at  one  end,  having  a  small  hole  in  it 
at  the  bottom.  A  bubble  should  come  to  rest  quickly,  but  should 
respond  easily  and  quickly  to  the  slightest  change  of  inclination 
of  the  tube. 

To  determine  the  radius  of  curvature  of  the  tube,  proceed  as 
follows:  Let  S  =  length  of  the  arc  over  which  the  bubble  moves 
for  an  inclination  of  1  second.  Let  R  =  its  radius  of  curvature. 
Then  S:  27rR  ::  1"  :  360°. 

From   which    R  =  206265x8     Or  =  S  = 

20b2bo 

S  may  be  found  by  trial,  the  level  being  attached  to  a  finely  divided 
circle.  Or,  bring  the  bubble  to  the  center  and  sight  to  a  divided 
rod;  raise  or  lower  one  end  of  the  level  and  again  sight  upon  the 
rod.  Call  the  difference  of  the  readings  />,  the  distance  of  the  rod 
fl,  and  the  space  which  the  bubble  moved  S.  Then  from  approx- 
imately similar  triangles 

d  S 


EXAMPLE  FOR  PRACTICE. 

1.  At  100  feet  distant,  the  difference  of  readings  was  0.02 
foot,  and  the  bubble  moved  0,01  foot,  What  is  the  radius  of  the 
bubble  tube  ?  Ans.  50  feetf 


PLANE  SURVEYING  29 

Locke's  Hand  Level.  This  instrument  consists  of  a  brass  tube 
six  inches  long  with  a  small  level  mounted  on  its  top  at  one  side 
of  the  center  near  the  object  end.  See  Fig.  25.  Underneath  the 
level  is  an  aperture  across  which  is  stretched  a  horizontal  wire 
attached  to  a  frame.  This  frame  is  made  adjustable  by  a  screw 
and  a  spring  working  against  each  other,  or  by  two  opposing 
screws  placed  at  the  ends  of  the  level  mounting.  In  the  tube, 
directly  below  the  level,  and  at  45°  to  the  line  of  sight,  is  placed 
a  totally  reflecting  prism  acting  as  a  mirror.  The  images  of  the 
bubble  and  wire  are  thus  reflected  to  the  eye.  The  prism  divides 


the  section  of  the  tube  into  two  halves,  in  one  of  which  is  seen 
the  bubble  and  wire  focussed  sharply  by  a  convex  lens  placed  in 
the  draw  tube  at  the  eye  end  of  the  instrument,  while  the  other 
permits  of  an  open  view.  Putting  the  instrument  to  the  eye  and 
raising  and  lowering  the  object  end  until  the  bubble  is  bisected  by 
the  horizontal  wire,  natural  objects  in  the  field  of  view  can  be  seen 
through  the  open  half  at  the  same  time,  and  approximate  levels 
can  then  be  taken.  To  prevent  dust  and  dampness  from  entering 
the  main  tube,  both  the  object  and  the  eye  ends  are  closed  with 
plain  glass. 

There  are  two  adjustments  necessary  in  this  instrument:  First, 
the  bubble  tube:  it  should  be  so  adjusted  that  the  bubble  will  be  in 
the  center  of  the  tube  when  the  instrument  is  horizontal.  Second, 
the  horizontal  wire;  it  should  bisect  the  bubble  when  the  latter  is 
in  the  center  of  the  tube.  The  methods  of  executing  these  adjust- 
ments are  so  apparent  it  will  be  unnecessary  to  dwell  upon  them 
here. 

The  instrument  is  intended  to  be  carried  in  the  pocket  and  is 
of  especial  value  upon  reconnaissance  surveys,  and  for  sketching  in 
topography  upon  preliminary  surveys. 

For  topographical  purposes,  the  topographer  should  provide 
a  rod  about  eight  feet  long,  divided  into  foot  lengths,  the  divisions 


30  PLANE  SURVEYING 

painted  alternately  red  and  white.  Upon  this  rod,  the  topographer 
should  mark  by  a  notch  or  other  means,  the  height  of  his  eye  above 
the  ground.  Standing  then  upon  a  station  of  the  line  of  survey, 
the  topographer  directs  his  assistant  to  carry  the  rod  out  upon 
either  side  of  the  line  and  in  a  direction  at  right  angles  thereto, 
until  a  point  having  the  proper  elevation  above  or  below  the  center 
line  is  found  as  determined  by  the  topographer  holding  the  instru- 
ment in  a  horizontal  position  at  the  eye.  The  topographer  then 
paces  the  distance,  while  the  assistant  carries  the  rod  to  the  next 
point.  It  is  evident  that  if  the  line  of  sight  from  the  instrument 
coincides  with  the  mark  upon  the  rod,  the  two  points  upon  the 
ground  are  at  the  same  level.  If  the  line  of  sight  strikes  the  rod, 
say  one  foot  below  the  mark  upon  the  rod,  it  is  evident  that  the 
ground  where  the  rod  is  held  is  one  foot  higher  than  \vhere  the 
instrument  is  held.  These  operations  can  be  repeated  indefinitely 
and  made  to  extend  as  far  as  necessary  upon  either  side  of  the  line. 
The  points  of  proper  and  equal  elevation  are  then  connected  form- 
ing contour  lines,  but  the  topographer  should  fill  in  details  by  the 
eye.  The  methods  of  keeping  the  field  notes  will  be  illustrated 
and  described  later. 

Let  BCXDEFG  and  H,  Fig.  26,  represent  the  successive  rod 
readings  on  the  right  of  the  center  line  A,  and  B'  C'  D'  E'  F'  G'  the 
readings  on  the  left.  Now  suppose  the  leveler  stands  with  a  Locke- 
level  at  zero  and  the  rod  is  held  vertically  at  B.  The  line  of  sight 
ab  bisects  the  rod  at  8.6  feet.  The  distance  from  the  ground  to  the 
observer's  eye  is  5.5  feet.  Thus  it  is  apparent  the  elevation  at  B 
will  be  3.1  feet  lower  than  at  A.  The  observer  now  paces  the  dis- 
tance between  A  and  B,  and  finds  it  to  be  50  feet.  The  reading  is 
now  taken  at  C  on  the  line  of  sight  cd  which  reads  6.2  feet,  hence 
the  elevation  of  C  is  .7  foot  lower  than  B,  and  the  distance  be- 
tween 20  feet.  Suppose  an  attempt  is  made  to  take  a  reading  near 
D.  Since  the  horizontal  plane  from  the  observer's  eye  to  the 
ground  does  not  strike  the  rod,  it  is  apparent  that  the  rod  is  too 
faraway,  therefore  it  should  be  moved  back^to  a  point  X  where  the 
horizontal  plane  ?f  will  bisect  the  rod  at  some  division.  The  ele- 
vation of  X  having  been  ascertained,  pace  the  distance  CX,  as  in 
the  former  cases.  This  method  is  continued  until  II  is  reached, 
taking  the  rod  readings  at  g  A,  ij,  k  I,  m  u,  and  s  y  and  pacing 


38 


PLANE  SURVEYIKG 


31 


the  distance  between  each.  The  same 
method  is  used  on  the  left-hand  side  of 
the  center  line.  However,  where  the 
surface  of  the  ground  has  an  abrupt 
change  between  stations,  it  is  customary 
to  take  cross  sections  at  such  changes  and 
ascertain  the  distances  between  the  sta- 
tions by  pacing;  the  center  line  at  such 
points  is  accepted  as  zero  ;  in  the  same 
manner  perform  the  operation  as  if  at 
a  station.  Where  a  cross  road  inter- 
sects the  center  line  or  any  portion  of  the 
cross  section,  take  readings  at  places  that 
show  an  abrupt  change,  as  the  top  of  a 
bank,  side  of  the  road,  or  gutter,  center  ^  \ 

of  the  road  and  on  the  other  side  in  the 
same  way  and  place  as  before.  This  rule 
holds  good  in  places  where  small  streams 
are  situated.  It  is  not  necessary  to  find 
the  depth  of  the  water,  because  the  pur- 
pose of  the  cross  section  deals  solely  with 

J  CENTER  LINE 

the  surface.  Where  obstacles  prevent  the 
section  being  run  at  right  angles  to  the 
center  line,  use  the  method  of  off -sets  and 
secure  the  desired  elevation  as  closely  ap- 
proximate as  circumstances  will  permit. 

The  Abney  Hand- Level  and  Clino- 
meter. This  instrument  is  similar  to  the 
Locke  hand-level,  see  Fig.  27,  but  the 
small  spirit  level  mounted  on  top  can  be 
moved  in  the  vertical  plane  and  is  clamped 
to  a  dial  graduated  upon  one  side  into 
single  degrees  and  upon  the  other  into 
slope  ratios,  so  that  it  is  possible  to  meas- 
ure angles  of  slope. 

The  adjustments  of  the  instrument  are 
the  same  as  for  the  Locke  hand-level. 

The  instrument  can  be  used  in  the  field 


PLANE  SURVEYING 


in  the  same  manner  as  the  Locke  hand-level,  but  is  of  more  universal 
application.  It  is  of  especial  value  upon  steep  slopes  when  the  effi- 
ciency of  the  Locke  level  would  be  limited  by  the  length  of  rod. 
In  using  the  Abney  instrument  it  is  only  necessary  to  mark  the 
height  of  the  eyes  upon  the  rod.  In  sighting  upon  the  rod,  with 
the  horizontal  line  coinciding  with  the  mark  upon  the  rod,  move  the 
vertical  circle  until  the  bubble  is  in  the  center  of  the  tube.  Read 
the  vertical  angle,  and  the  tangent  of  this  angle  multiplied  by  the 
horizontal  distance  to  the  rod  will  give  the  difference  of  elevation. 
If  the  distance  to  the  rod  is  measured  along  the  slope  of  the 


ground,  multiply  this  distance  by  the  sine  of  the  vertical  ano-le  to 
get  the  difference  of  elevation. 

The  most  satisfactory  method  of  using  this  instrument  is  in 
connection  with  a  straight  edge  from  8  to  10  feet  in  length.  The 
straight  edge  is  laid  upon  the  ground  parallel  to  the  direction  of 
slope  and  the  clinometer  is  then  applied  to  it,  the  vertical  circle 
being  turned  till  the  bubble  is  in  the  center.  The  angle  of  slope 
is  then  read,  or  better  still,  the  slope  ratio  is  read  from  the  vertical 
circle.  This  operation  is  repeated  at  every  change  of  slope,  the 
distances  being  either  paced  or  measured  with  a  tape.  For  in- 
stance, suppose  the  slope  is  found  to  be  GO  feet  in  length  and  the 
slope  ratio  as  given  by  the  clinometer  is  ^.  It  is  evident  then 
that  at  the  end  of  the  slope  the  difference  in  elevation  will  be  6 


40 


PLANE  SURVEYING 


33 


feet.  The  instrument  is  sometimes  fitted 
with  a  small  compass  and  a  socket  for  use 
upon  a  tripod  or  Jacob  staff. 

The  Leveling  Rod  is  an  important  part  of 
the  leveling  outfit;  it  is  used  in  measuring 
the  vertical  distance  between  the  horizontal 
plane  through  the  line  of  sight  and  the  point 
upon  which  the  rod  is  held.  There  are  three 
forms  in  common  use  known  as  the  New 
York,  Philadelphia  and  Boston.  They  are 
•made  of  hard  wood  6^  feet  long,  sliding  out 
to  12  feet  and  provided  with  target,  vernier 
and  clamps. 

Leveling  rods  are  of  two  kinds,  the  target 
and  the  self -reading.  Of  the  target  rods, 
the  New  York  and  Boston  are  generally  used 
for  precise  work.  Of  the  self-reading  rods, 
the  Philadelphia  shown  in  Fig.  28  is  in  more 
common  use.  The  self-reading  rods  are  used 
only  in  connection  with  that  class  of  work 
where  approximate  accuracy  only  is  required; 
this  form  is  generally  read  to  hundredths  of  a 
foot  and  can  be  read  directly  from  the  instru- 
ment by  the  observer  without  the  aid  of  the  tar- 
get, as  is  suggested  by  the  name.  However, 
with  the  aid  of  the  target  this  rod  can  be  read 
to  thousandths  of  a  foot  approximately.  The 
target  is  used  when  greater  accuracy  is  re- 
quired and  when  the  rod  is  so  far  from  the 
instrument  that  it  cannot  be  distinctly  read. 

The  rod  consists  of  a  graduated  scale  di- 
vided into  feet,  tenths  and  hundredths  of  a 
foot,  and  when  properly  made,  readings  to 
thousandths  of  a  foot  can  be  easily  taken. 
Fig  28  The  numbers  making  the  tenths  should  be 

0.06  foot  long  and  so  placed  that  one-half  the 

length  is  above  and  one-half  below  the  line.  The  numbers  marking 
the  feet  are  0.10  foot  long  and  each  is  bisected  by  the  foot  mark. 


41 


34 


PLANE  SUEVEYING 


This  class  of  rod  is  painted  white,  the  foot  graduations  are  red  and 
the  tenths  and  hundred ths  are  black  horizontal  lines. 

No  attempt  will  be  made  to  describe  the  reading  of  the  vernier 
of  either  the  New  York  or  Boston  rod, 
but  the  Philadelphia  rod  is  so  divided  as 
to  make  its  reading  easily  understood. 
With  this  rod  each  side  of  the  black 
horizontal  line  indicates  lOOths,  that  is, 
the  lower  side  of  the  first  black  space 
is  called  "one,"  and  the  upper  side  of 
the  same  space  is  called  "two,"  the 
lower  side  of  the  third  space  is  called 
"three"  and  so  on  until  the  tenth  is 
read. 

The  reading  is  taken  without  the  aid 
of  the  target,  in  feet,  tenths  and  hun- 
dredths  as  the  case  may  be.  The  mov- 
able target  has  a  vernier  which  reads 
to  thousandths  of  a  foot  and  is  read 
from  zero  to  ten.  To  read  this  rod, 
move  the  target  to  any  convenient  place 
on  the  scale  of  the  rod  and  note  where 
the  vernier  at  zero  coincides  with  a  black 
horizontal  line  ;  then  note  where  a  line 
of  the  vernier  coincides  with  a  line  of 
the  scale.  For  example,  if  the  zero  of 
the  vernier  is  just  above  one  foot,  four- 
tenths  and  five  hundredths,  as  shown 
in  Fig.  29,  and  a  line  of  the  graduation 
of  the  vernier  coincides  at  7  with  a  hori- 
zontal black  line  on  the  rod,  the  reading  will  be  1.457  as  is  shown 
in  Fig.  29.  If  reading  to  the  nearest  100th,  the  reading  will  be 
1.46.  This  is  because  the  7  naturally  brings  the  zero  .002  above 
the  line  of  graduation  on  the  rod,  therefore,  the  zero  of  the  vernier 
is  .002  nearer  the  6  than  the  5,  hence,  the  reading  is  as  above. 
Should  the  vernier  read  .002  instead  of  .007  the  reading  would  be 
1.45.  It  is  apparent  that  .002  now  brings  the  zero  of  the  vernier 
below  the  line,  hence  it  will  be  nearer  to  the  5  than  the  6,  thus  the 


Fig,  29. 


LEVELER  AND  RODMAN  ENGAGED  IN  BASCULE  BRIDGE  CONSTRUCTION  OVER 
CHICAGO  RIVER,  CHICAGO,  ILLINOIS 


PLANE  SURVEYING 


35 


rod  reading  is  1.45.  Therefore,  in  all  readings  with  the  Phila- 
delphia rod,  read  the  thousandths  to  the  nearest  half  hundredth. 
This  is  true  whether  or  not  the  lines  coincide. 

These  readings  apply  only  to  the  face  of  the  rod  or  to  6^  feet. 
When  the  rod  is  extended  to  12  feet,  or  any  fractional  part  thereof, 
the  reading  is  a  little  different,  both  as  to  its  graduation  and 
vernier.  The  scale,  of  course,  is  the  same  on  the  face  of  the  rod 
when  extended,  except  as  to  the  vernier,  which  is  placed  on  the 
back  at  6^  feet  and  the  scale 
of  graduation  on  the  ex- 
tended part  of  the  rod  is  also 
on  the  back  of  the  extension 
which  runs  through  the  ver- 
nier, as  shown  in  Fig.  80. 
The  scale  of  hundredths  is  the 
only  part  to  be  particularly  A> 
observed,  together  with  the 
vernier  in  the  former. 

For  example,  the  first 
horizontal  black  space  equals 
"one,"  which  is  the  top  line 
of  the  foot  mark.  The  lower 
side  of  the  first  black  space  is 
"two,"  and  the  upper  side  of 


Fig.  30. 


Fig.  31. 


the  same  space  is  "three"  hundredths,  and  so  on  until  the  tenth 
is  reached.  The  tenth  and  feet  are  placed  the  same  as  on  the  face 
of  the  rod.  The  vernier,  as  already  stated,  is  a  little  different  in 
point  of  reading  and  is  graduated  from  ten  to  zero,  instead  of  zero 
to  ten,  as  on  the  movable  target.  However,  with  some  recently- 
made  rods  of  this  type,  the  scale  and  vernier  reading  is  the  same 
throughout.  See  Fig.  31.  The  graduation  at  ten  is  taken  as  the 
zero  in  determining  thousandths.  The  vernier  in  question  is  firmly 
attached  to  the  upper  end  of  the  rod  6^  feet,  (and  the  extension  of 
the  rod  runs  through  this  vernier).  The  differences  in  graduation 
of  the  two  sides  should  be  carefully  noted.  The  rod  has  two 
clamp  screws,  one  attached  to  the  movable  target  and  the  other 
near  the  vernier  on  the  back  of  the  rod.  In  running  the  rod,  it 
is  customary,  where  a  target  or  rod  reading  exceeding  6i  feet  is 


43 


PLANE  SURVEYING 


desired,  to  set  the  target  at  6|  feet  and  run  the  rod  to  its  full  length, 
then  move  down  as  signalled;  where  no  target  reading  is  required, 
run  the  rod  to  its  full  extent  (12  feet)  and  as  the  face  of  the  rod  has 
a  scale  throughout,  the  reading  can  be  taken  from  the  instrument. 

Should  the  instrument  not  be  near 
enough  to  enable  the  leveler  to  see 
the  rod  distinctly  without  the  aid  of 
the  target,  he  should  first  read  the 
rod  through  the  telescope  of  the  in- 
strument and  then  notify  the  rod- 
man  at  what  distance  the  intersection 
of  the  cross-hairs  in  the  instrument 
approximately  bisects  the  rod,  such 
as  3.21,  which  means  three  feet,  two 
tenths  and  one  hundredth.  The  rod- 
man  then  sets  his  rod  to  read  this 
distance  and  another  sight  is  taken, 
being  careful  to  have  the  rod  plumb. 
Should  the  intersection  of  the  target 
fail  to  coincide  with  the  cross -hairs 
in  the  instrument,  the  leveler  then 
signals,  or  calls  out  if  sufficiently  near 
to  do  so,  the  true  rod  reading,  as  up 
a  tenth,  down  two  hundredth^,  as  the 
case  may  be,  and  the  target  is  placed  at  this  distance.  When  pre- 
cision is  required,  this  method  is  relied  upon  only  for  the  approx- 
imate placing  of  the  target;  the  method  used  in  this  case  is  to 
slowly  move  the  target  by  standing  behind  the  rod  and  holding 
it  between  the  thumb  and  fingers  of  one  hand,  while  the  target  is 
moved  with  the  other.  Then  the  target  is  slowly  moved  by  the 
signals  of  the  observer. 

When  a  slow  motion  with  the  hand  above  the  shoulder  or 
below  the  hip  is  made  by  the  observer,  it  means  that  the  rod 
is  to  be  moved  in  that  direction  a  fractional  part,  as  one  tenth, 
but  when  a  quick  motion  is  made  and  the  hand  drawn  back  in 
the  same  manner  it  implies  that  the  target  is  to  be  moved  just  a 
trifle.  In  this  way  and  by  proper  attention  to  the  signals  of  the 
observer,  the  rodman  can  become  an  efficient  and  helpful  assistant, 


Fig.  32. 


PLANE  SURVEYING 


37 


thereby  saving  mncli  time.  "When  the  target  is  finally  set,  the 
rodman  reads  the  rod  and  calls  out  the  reading  to  the  observer, 
when  within  reasonable  distance.  The  target  rods  are  read  en- 
tirely by  the  rodman,  and  the  readings  are  kept  by  him  in  a  note 
book  for  that  purpose ;  these  notes  should  be  given  to  the  observer 


at  every  opportunity  and  results  checked.  To  obtain  correct  re- 
suits  when  leveling,  it  is  absolutely  essential  that  the  rod  be  ver- 
tical  and  the  rodman  should  remember  to  hold  the  rod  in  this 
position. 


45 


38  PLANE  SURVEYING 

The  observer  or  leveler,  by  means  of  the  vertical  wire  upon 
the  target  of  an  ordinary  leveling  rod,  can  tell  whether  or  not  a 
rod  is  vertical  and  in  a  position  at  right  angles  to  the  line  of  sight, 
but  he  is  not  able  to  determine  whether  the  top  of  the  rod  is  in- 
clined towards  the  instrument  or  in  the  opposite  direction;  because 
when  looking  through  the  telescope  of  a  level  he  can  see  only  a 
fractional  part  of  the  rod.  Therefore  the  necessity  of  overcoming 
this  difficulty  led  to  the  invention  of  the  bent  target  which  obviates 
this  latter  trouble  as  can  readily  be  seen  from  Fig.  32.  The 
American  target  fulfills  the  same  requirements,  but  differs  from  the 
ordinary  target  in  having  two  discs,  one  behind  the  other,  as  in 
Fig.  33.  The  principle  of  construction  of  this  target  is  ex- 
tremely simple,  and  may  be  best  explained  in  the  figures  above 
Suppose  a  target  of  the  old  kind,  which  in  its  front  view  looks 
exactly  like  the  front  view  of  the  new  target  in  A,  to  be  cut 
along  the  vertical  lines  an,  ll>,  thus  dividing  it  into  three  parts; 
that  is,  one  center-piece  and  two  wings.  Suppose  furthermore, 
the  centerpiece  to  remain  in  its  former  place  at  the  front  of  the 
rod,  while  the  two  wings  are  removed  to  the  rear  of  the  rod.  Then 
the  result  evidently  will  be  that  the  horizontal  line  ec,  d<l,  will 
appear  as  one  unbroken  line  to  the  observer,  only  when  the  rod  is 
held  perfectly  vertical.  Any  deviation  either  towards  the  instru- 
ment, or  away  from  it  will  cause  the  two  parts  cc  and  dd  of  the 
horizontal  line  situated  at  the  rear  of  the  rod  in  the  wings  of  the 
target  to  show  either  above  or  below  that  part  al  of  the  horizontal 
line  which  is  situated  in  the  front  of  the  rod  in  the  centerpiece  of 
the  target. 

Whether  using  the  bent  target,  the  ordinary  target  or  no 
target  at  all,  it  is  apparent  that  the  rodman  should  hold  the  rod  in 
a  vertical  position,  known  as  plumb.  This  can  be  done  by  stand- 
ing directly  behind  the  rod  with  both  feet  together  or  apart,  as  the 
rays  of  the  sun  may  require,  governing  shadows,  and  holding  the 
rod  between  the  thumb  and  finger  of  one  hand  while  moving  the 
target  with  the  other.  After  the  target  is  set,  both  hands  are 
brought  in  line  with  the  shoulders,  and  standing  erect,  the  hand 
should  touch  the  rod  very  lightly,  so  that  it  will  almost  stand  in  a 
vertical  position  by  itself;  or  when  standing  in  that  position  if  the 
center  of  the  rod  or  the  corner  is  made  to  coincide  perpendicularly 


46 


PLANE  SURVEYING  39 

with  the  nose  and  chin,  it  will  be  plumb.  The  rodman  should 
never  put  his  hands  around  the  rod. 

Another  way  is  to  sight  along  the  line  of  some  buildin^ 
apparently  in  line.  There  are,  of  course,  many  different  methods 
of  signalling,  but  the  ones  mentioned  are  frequently  used. 

When  the  rod  is  to  be  read  without  the  aid  of  a  target  or  with 
the  ordinary  target,  it  frequently  happens  that  the  rod  is  not  ver- 
tical and  the  signal  used  for  bringing  it  in  its  proper  vertical 
plane  is  the  raising  of  either  the  right  or  left  hand  in  a  vertical 
position,  which  indicates  that  the  rod  should  be  inclined  in  that 
direction.  In  so  doing  move  the  rod  slowly  until  the  hand  is  low- 
ered. After  the  target  has  been  set  in  its  proper  position,  clamp  it 
by  the  screw  on  its  side,  then  give  another  sight  and  note  the  sig- 
nals of  the  observer. 

If  the  target  is  to  be  moved,  the  observer  sliould  hold  the  palm 
of  one  hand  in  the  direction  the  target  is  to  be  moved.  The  observer 
should  use  but  one  hand  in  signalling  the  rod;  if  the  target  is  to 
be  lowered,  he  should  hold  his  hand  below  his  hip,  palm  down. 
To  raise  the  target  he  should  hold  his  hands  above  his  shoulder, 
palm  up.  Any  considerable  change  in  the  position  of  the  target 
is  denoted  by  a  more  or  less  violent  motion  of  the  hand.  If  a  very 
slight  change  is  desired,  the  observer  should  liold  his  hand  in  the 
proper  position  without  moving  it  up  or  down.  When  the  proper 
position  of  the  target  has  been  obtained,  the  observer  indicates  the 
fact  by  raising  both  arms  above  the  head  and  moving  them  in  the 
arc  of  a  circle  to  indicate  that  the  rod  reading  at  that  particular 
place  is  complete. 

New  York  Rod.  This  rod  resembles  the  Philadelphia  rod  as 
to  its  use  and  dimensions,  but  differs  as  to  scale  and  vernier  read- 
ing. The  scale  is  divided  into  feet,  tenths  and  hundredths,  the 
same  as  the  Philadelphia  rod,  except  the  graduations  of  the  hun- 
dredths, which  instead  of  having  the  sides  of  one  black  space  each 
equivalent  to  0.01  foot,  as  on  the  Philadelphia  rod,  the  hundredths 
are  distinct  by  themselves,  therefore,  each  line  between  the  tenth 
is  .01  part  of  the  scale.  Fig.  3-4  shows  the  rod  at  its  full  length, 
and  Fig.  35  shows  a  sectional  part  thereof  with  its  movable  target 
set  at  6-J  feet  and  the  black  horizontal  lines  each  indicate  .01  as 
more  fully  explained  hereafter.  Fig.  36  shows  the  side  with  its 
graduations,  as  on  the  face  of  the  rod  and  is  set  at  6^  feet. 


47 


40 


.  PLANE  SURVEYING 


The  rod  Las  a  movable  target  which  carries  a  vernier,  and 
allows  readings  to  thousandths.  It  cannot,  however,  be  read  with- 
out the  aid  of  the  target  and  is  used  for  the  most  part, 
where  precision  is  required;  it  probably  commends  its- 
self  to  a  greater  number  of  engineers,  because  of  its 
stiffness  and  wearing  qualities.  For  elevations  up  to 
6^  feet,  the  target  is  used  in  the  same  manner  as  the 
former  /od  by  sliding  it  up  or  down  upon  the  rod. 
Above  0^  feet,  as  with  all  the  rods,  the  target  is  clamped 


FiS-  34-  Fig.  35.  Fig.  36. 

at  the  GA-foot  division.  The  back  of  the  rod  slides  upon  the  front 
half  and  when  so  extended  the  vernier  is  on  each  of  the  narrow 
sides,  impressed  in  the  wood  (See  Fig.  36).  It  should  be  noted 
that  while  the  vernier  on  the  Philadelphia  rod  is  situated  on  the 


48 


PLANE  SURVEYING  41 

back  when  extended,  with  the  New  York  rod  it  is  on  its  narrow 
sides.  The  vernier  in  question  is  somewhat  different  from  the 
one  found  on  the  Philadelphia  rod,  since  it  is  provided  with  a 
direct  vernier,  while  the  other  is  provided  with  an  indirect  vernier. 
The  former,  as  has  been  explained,  is  usually  placed  below  the 
center  of  the  target,  that  is,  the  zero  is  placed  below  the  in- 
tersection of  the  horizontal  and  vertical  lines  of  the  target. 
In  almost  every  case  this  causes  confusion  because  the  rodnian 
has  been  taught,  by  reason  of  using  a  Philadelphia  target  rod, 
to  read  the  scale  at  the  zero  of  the  vernier  to  tho  fractional 
parts  of  a  foot  by  looking  along  the  vernier  for  the  coincid- 
ing lines.  There  need  be  very  little  confusion  in  reading  the  New 
York  rod,  if  it  is  remembered  that  the  center  of  the  target  is  set 
by  the  leveler  and  not  the  zero  of  the  vernier  as  on  the  Philadel- 
phia rod.  '  Careful  observation  of  the  vernier  will  show  that  the 
zero  of  the  vernier  is  placed  at  the  intersection  of  the  horizontal 
and  vertical  lines  of  the  target.  The  method  of  using  the  clamps, 
setting  the  target,  etc.,  is  the  same  as  that  of  the  Philadelphia  rod. 

The  Boston  Rod  is  made  of  mahogany,  is  of  the  same  length 
and  slides  out  as  the  rods  just  described.  It  is  distinctly  a  target 
rod  and  cannot  be  read  without  its  aid.  The  scale  and  vernier,  how- 
ever, are  on  the  narrow  sides  and  can  be  read  to  thousandths  or 
any  fractional  part  of  a  foot.  The  target  is  fixed  upon  one-half  of 
the  rod  for  elevations  less  than  6^  feet.  The  target  end  is  held  upon 
the  ground  and  the  front  of  the  rod  slides  upon  the  back,  as  shown 
in  Fig.  37.  Above  6^  feet  the  rod  is  inverted  as  shown  in  Fig. 
38,  and  is  then  used  in  much  the  same  way  as  the  New  York  rod. 
The  figures  above  referred  to  show  the  sides  of  the  rod  with  its  scale 
upon  it.  The  screws  at  each  end  act  as  clamps.  In  the  old  style  of 
Boston  rod  a  wooden  target  was  screwed  to  the  rod  with  the  result 
that  the  target  would  warp  and  twist  or  be  knocked  off,  thus  render- 
ing  the  rod  useless.  The  best  type  of  Boston  rod  is  fitted  with  a 
bent  metal  target.  This  form  is  very  serviceable  and  satisfactory. 
It  will  be  apparent  from  the  foregoing  description  that  the  rod  is 
read  altogether  by  vernier,  the  scales  and  vernier  being  on  the 
side.  It  is  the  lightest  and  neatest  rod  of  the  three  but  the  least 
used. 

Cross-Section  Rod.     This  is  a  rod  10  feet  in  length,  painted 


42 


PLANE  SURVEYING 


E-5 


white,  with  black  graduations;  it  is  divided  into  feet,  tenths  and 
hundredths,  (See  Fig.  39).  The  scale  is  on  both  sides.  At  each 
end  is  a  spirit  level  bubble  with  graduations  on  the  upper  side  of 
the  tube  to  bring  the  rod  in  a  horizontal  plane.  In  the  center  of 
the  rod  is  an  opening  for  the  hand,  and  thereby  it  can  be  easily 

taken  from  place  to  place. 
The  purpose  of  this  rod  is  to 
simplify  the  lengthy  calculations 
in  taking  cross  sections;  this 
will  be  more  fully  explained 
under  its  respective  head. 

In  Fig.  39  A  and  B  are  the 
bubbles.  It  is  apparent  that  if 
one  end  of  the  rod  is  placed  on 
the  side  of  a  hill  and  the  other 
raised  in  a  vertical  position  until 
the  bubble  appears  in  the  center 
of  the  tube,  the  base  of  the  rod 
will  be  in  a  horizontal  plane. 

Ranging  Poles.  Fig.  40 
shows  the  three  forms  of  rang- 
ing poles  (called  flags)  in  com- 
mon use,  all  of  which  are  from 
C  to  10  feet  in  length,  made  of 
hardwood,  octagonal  in  shape; 
they  are  tapered  from  the  top 
down  and  each  foot  painted  al- 
ternately red  and  white,  and  pro- 
Fig.  37.  Fig.  38.  vided  with  steel  shoes,  except 
the  smallest  one,  which  consists  of  an  iron  tubular  rod  ^-inches  in 
diameter  and  used  for  the  most  part  on  construction  work.  These 
flags  are  for  the  purpose  of  establishing  points  or  retaining  a  given 
line  indefinitely;  it  is  an  important  tool  of  the  surveyor's  outfit. 
To  use  this  flag,  it  is  placed  approximately  at  some  reasonable 
distance  from  the  instrument  and  then  by  the  signals  of  the 
observer  is  moved  until  the  line  of  sight  through  the  instrument 
bisects  it.  It  should  be  held  in  a  vertical  position  and  governed 
by  the  same  methods  used  in  placing  a  leveling  rod  in  a  vertical 


50 


PLANE  SURVEYING 


43 


position.  It  is  also  a  convenient  device  for  measuring,  approx- 
imately, distances  not  exceeding  six  feet,  but  where  any  great 
amount  of  accuracy  is  desired,  the  method  should  not  be  relied 
upon.  It  is  an  advantage,  however,  to  use  the  flag  as  a  check, 


I'1'1'1 


when  it  may  appear  that  some  discrepancy  has  occurred. 

INSTRUMENTS. 

The  Wye  Level.  There  are  three  kinds  of  leveling  instru- 
ments in  common  use,  viz:  The  "Wye  level  with  four  leveling 
screws,  the  Wye  level  with  three  leveling  screws  and 
P  Q  fi  the  Dumpy  level.  The  Wye  level  derives  its  name 
from  the  vertical  forked  arms,  called  Wyes,  in  which 
the  telescope  rests.  It  is  clamped  to  them  by  collars 
which  may  be  raised  allowing  the  telescope  to  be 
turned  on  its  horizontal  axis  or  lifted  out  entirely. 
It  is  also  referred  to  as  the  four-screw  level.  Like 
other  levels  it  is  used  for  the  purpose  of  ascertaining 
a  horizontal  line  of  sight  parallel  to  a  spirit  level 
and  perpendicular  to  the  vertical  axis.  The  line  of 
sight  is  fixed  in  the  telescope  by  the  intersection  of 
cross-hairs.  A  spirit  level  is  attached  to  the  under 
side  of  the  telescope  and  is  protected  except  on  top 
by  a  metal  tube.  In  the  barrel  of  the  telescope  slide 
tv,T6  tubes,  in  one  of  which  is  an  eye-piece;  in  the 
other  is  the  objective. 

The  eye-piece  usually  found  with  the  four-screw 
leveling  instrument  is  of  the  erecting  type.     The  in- 
verting eye-piece  as  distinguished  from  the  erecting 
eye-piece  has  two  lenses  instead  of  four.     The  result 
is  that  the  inverting  eye-piece  permits  more  light  to 
reach  the  eye  of  the  observer,  and  is  therefore  better 
adapted  to  precise  leveling.     At  first  some  inconven- 
Fig.  40.        ience  is  experienced  by  the  fact  that  all  objects  are 
upside  down,  but  a  little  experience  will  soon  con- 
vince the  observer  that  all   the  advantages   whether  for  a  four- 


51 


44 


PLANE  SURVEYING 


screw  instrument  or  a  three-screw  instrument,  lie  with  the 
inverting  eye-piece.  Nearly  all  makers  give  a  purchaser  his  choice 
of  the  style  of  eye-piece  without  extra  charge.  In  purchasing  an 


instrument  it  should  be  noticed  whether  the  eye-piece  is  adjusted 
by  a  straight  pull  or  by  a  spiral  motion,  because  the  spiral  motion 
is  usually  considered  more  satisfactory. 


52 


PLANE  SURVEYING 


45 


WYE   LEVEL — THREE   SCREW. 


Fig.  41.      DUMPY   LEVEL. 


PLANE  SURVEYING 


An  inexperienced  observer  upon  looking  through  the  level 
and  finding  no  cross-hairs  may  suspect  "that  the  cross-hairs  are 
broken.  It  must  not  be  forgotten  that  the  eye- 
piece  must  be  focused  before  the  cross-hairs  will 
come  into  the  range  of  vision.  Having  once 
focused  the  eye-piece  upon  the  cross-hairs,  the 
adjustment  will  stand  for  a  long  time  if  the  eye- 
piece is  undisturbed. 

The  object  glass  is  moved  in  and  out  by  means 
of  a  pinion  which  works  on  a  rack  attached  to  a 
sliding  tube  and  moves  in  the  axis  of  the  barrel, 
passing  through  the  run  which  is  inclined  in 
the  barrel.  The  instrument  is  provided  with  a 
clamp,  slow  motion  and  leveling  screws  and 
mounted  on  a  tripod.  The  two  former  screws 
are  situated  directly  under  the  horizontal  bar 
and  revolve  with  the  telescope. 

The  Line  of  Collimation  of  a  level  is  the  line 
joining  the  optical  center  of  the  object-glass 
and  the  intersection  of  the  cross-hairs,  and  since 
this  line  determines  the  point  towards  which  the 
telescope  is  directed,  it  should  coincide  with  the 
optical  axis  of  the  telescope.  The  eye-piece 
and  object-glass  must  be  accurately  centered. 

Instrumental  Parallax  is  an  important  con- 
dition of  focusing  due  to  the  fact  that  the  image 
does  not  fall  in  the  plane  of  the  cross-hairs. 

To  determine  this,  direct  the  telescope  upon  an 
object  and  focus  the  eye-piece  so  that  tho  cross- 
hairs are  perfectly  distinct.  Then  turn  the  tele- 
scope upon  the  object  which  is  to  be  observed, 
and  focus  the  object  glass  until  the  image  is 
clearly  defined.  Move  the  eye  from  side  to  side 
and  note  whether  there  is  any  apparent  move- 
ment of  the  cross-hairs  and  image.  If  any  is 
seen,  the  two  operations  are  to  be  repeated  until 
all  parallax  is  removed. 
This  adjustment  depends  upon  the  eye  of  the  observer  and 
when  made  for  one  person  may  not  be  correct  for  another. 


PLANE  SURVEYING  47 


Spherical  Aberration.  This  defect  is  caused  by  combining 
lenses  of  different  curvatures  so  that  objects  on  the  side  of  a 
field  of  view  are  seen  less  distinctly  than  those  in  the  center.  To 
test  the  object  glass  for  this  defect  cover  the  outer  edge  with  an 
annular  ring  of  paper  and  focus  upon  some  desired  object.  Then 
remove  the  ring  and  cover  the  central  spot  of  the  glajss;  if  no 
change  of  focus  is  needed  the  glass  has  no  spherical  aberration. 

To  test  the  eye-piece,  sight  to  a  heavy  black  line  drawn  on 
white  paper  and  held  near  the  side  of  the  field  of  view.  If  it  ap- 
pears perfectly  straight  the  eye-glass  is  a  good  one. 

Chromatic  Aberration  is  a  defect  caused  by  combining 
lenses  of  different  and  improper  varieties  of  glass  so  that  the  yel- 
low or  purple  colors  appear  on  the  edge  of  the  field.  To  test  the 
telescope  for  this  defect  focus  it  upon  a  bright  distant  spot  and 
slowly  move  the  object  glass  out  and  in.  If  no  colors  are  observ- 
ed around  the  edge  of  the  field  of  view  the  telescope  is  free  from 
this  defect. 

Adjustments.  The  adjustments  of  the  Wye-level  are  three 
in  number  and  should  be  made  in  the  following  order: 

1.  To  make  the  line  of  collimation  parallel  to  the  bottoms 
of  the  collars. 

2.  To  make    the  axis  of  the  bubble  tube  parallel  to  the  line 
of  collimation. 

3.  To  make  the  axis  of  the  bubble  tube  perpendicular  to  the 
vertical  axis  of  the  instrument. 

To  make  the  test  for  the  first  adjustment  set  up  the  instru- 
ment firmly  upon  solid  ground,  shaded  from  sun  and  wind.  Di- 
rect the  telescope  towards  the  side  of  a  building,  a  fence  or  other- 
convenient  object  and  carefully  center  the  intersection  of  the 
cross-hairs  upon  a  well-defined  point,  such  as  the  head  of  a  tack. 
Clamp  the  vertical  axis  and  loosen  the  telescope  clips.  Now 
slowly  revolve  the  telescope  in  the  wyes  and  note  if  the  intersec- 
tion of  the  cross-hairs  continues  to  cover  the  point.  If  so,  the 
line  of  collimation  is  in  adjustment. 

If  the  intersection  of  the.  cross -hairs  moves  off  the  point,  re- 
volve the  telescope  in  the  wyes  as  nearly  as  possible  through  180 
degrees  and  carefully  center  a  point  at  the  intersection  of  the 
cross-hairs  in  this  last  position.  Bisect  the  distance  between  the 


PLANE  SURVEYING 


two  points  and  establish  a  third  point:  by  means  of  the  screws  at- 
tached to  the  cross-hair  diaphragm,  move  the  diaphragm  so  that  the 
intersection  of  the  cross-hairs  covers  the  third  point.  Now  re- 
peat the  test  and  correct  the  position  of  the  cross-hair  diaphragm 
until  the  intersection  of  the  cross-hairs  covers  one  point  as  the 
telescope  is  revolved  in  the  wyes. 

The  horizontal  cross-hair  should  at  all  points  be  at  the  same 
distance  from  the  bottoms  of  the  collars.  To  test  this,  carefully 
center  one  extremity  of  the  hair  upon  a  point,  and  by  means  of 
the  tangent  screw,  slowly  revolve  the  telescope  upon  the  vertical 
axis,  and  if  the  hair  covers  the  point  from  end  to  end,  the  adjust- 
ment is  complete.  If  it  does  not,  the  hair  is  to  be  adjusted  by 
the  same  screws  as  before.  Making  this  adjustment  will  probably 
disturb  the  former  one,  and  the  two  are  to  be  repeated  in  succes- 
sion until  satisfactory. 

It  will  be  noticed  that  to  make  this  adjustment,  it  is  not  nec- 
essary to  level  the  instrument. 

Adjustment  of  the  Axis  of  the  Bubble-tube.  To  test  this 
adjustment,  first  throw  back  the  clips  holding  the  telescope  in  the 
wyes,  and  then  revolve  the  telescope  upon  the  vertical  axis 
to  bring  it  directly  over  a  pair^of  leveling  screws  and  clamp  the 
axis  firmly.  By  means  of  these  leveling  screws  bring  the  bubble 
to  the  center  of  the  tube  as  accurately  as  possible.  Now  without 
disturbing  the  instrument,  carefully  lift  the  telescope  out  of  the 
wyes  and  turn  it  end  for  end,  being  careful  when  replaced  in  the 
wyes  that  the  telescope  comes  to  its  seat  at  each  end.  If  the  bub- 
ble in  this  new  position  of  the  telescope  comes  to  rest  at  the  cen- 
ter of  the  tube,  the  axis  of  the  tube  is  in  adjustment. 

If  the  bubble  does  not  return  to  the  center,  bring  it  one-half 
way  back  to  the  center  by  means  of  the  vertical  screws  at  one  end 
of  the  bubble-tube,  and  the  remainder  of  the  way  by  the  two  lev- 
eling screws.  Now  repeat  the  test  and  correction  as  often  as  nec- 
essary until  the  bubble  remains  in  the  center  of  the  tube. 

Having  adjusted  the  bubble-tube  over  one  pair  of  screws, 
test  it  over  .he  other  pair. 

When  the  horizontal  hair  is  truly  horizontal,  the  line  of  colli- 
mation  and  the  axis  of  the  bubble-tube  should  be  iu  the  same 
vertical  plane.  To  test  this,  after  the  previous  adjustment  has 


56 


PLANE  SURVEYING  49 

been  completed,  loosen  the  clips  of  the  wyes  and  bring  the  bubble 
carefully  to  the  center  of  the  tube.  Now  slowly  revolve  the  tele- 
scope in  the  wyes  and  note  if  the  bubble  still  remains  in  the  cen- 
ter of  the  tube.  If  it  does,  the  line  of  colliraation  and  the  axis  of 
the  tube  are  in  the  same  plane.  If  the  bubble  runs  to  one  end  of 
the  tube,  bring  it  back  by  the  horizontal  screws  attached  to  one 
end  of  the  tube. 

It  must  be  borne  in  mind  that  the  first  and  second  adjust- 
ments must  be  carefully  made  and  that  they  are  absolutely  essen- 
tial if  satisfactory  results  are  to  be  attained  with  the  instrument. 

Adjustment  of  the  Vertical  Axis.  This  adjustment  is  not 
absolutely  essential,  provided  that  every  time  a  reading  is  taken 
the  bubble  is  brought  to  the  center  of  the  tube  by  means  of  the 
parallel  plate  screws.  However,  the  adjustment  will  expedite 
field  work  and  should  always  be  made. 

To  test,  level  the  instrument  carefully  over  both  pairs  of 
screws;  if  the  bubble  remains  in  the  center  of  the  tube  as  the  tele- 
scope is  revolved  on  the  vertical  axis  all  the  way  round,  the  ad- 
justment is  complete.  If  the  bubble  runs  to  one  end  as  the  tele- 
scope is  thus  revolved,  the  vertical  axis  is  out  of  adjustment  an<( 
may  be  corrected  as  follows.  Bring  the  telescope  directly  over  a 
pair  of  opposite  plate  screws,  and  by  means  of  these  screws  bring 
the  bubble  accurately  to  the  center  of  the  tube.  Now  revolve  the 
telescope  on  the  vertical  axis  as  nearly  as  possible  through  180 
degrees  and  note  the  displacement  of  the  bubble:  bring  the  bub- 
ble one-half  of  the  way  back  to  the  center  by  the  screws,  at  one 
end  of  the  bubble-bar  attached  to  the  wyes,  and  the  remainder  of 
the  distance  by  the  parallel  plate  screws.  .Repeat  the  test  and 
adjustment  until  the  bubble  remains  in  the  center  of  the  tube  in 
all  positions  of  the  telescope. 

Replacing  the  Cross-hairs.  The  cross-hairs  in  leveling  in- 
struments may  be  either  spider  webs  or  platinum  wire.  Spider 
webs  are  better,  but  in  selecting  them  care  should  be  taken  to  see 
that  they  are  free  from  dust  and  dampness.  Probably  the  web  of 
the  little  black  spider  is  the  most  satisfactory,  but  the  wob  of  the 
common  spider  will  give  good  results,  especially  if  freshly 


57 


50  PLANE  SURVEYING 


spun.  Spider  webs  can  best  be  carried  by  winding  them 
around  a  stick. 

The  cross-hairs  are  attached  to  the  small  diaphragm  set  at 
the  principal  focus  of  the  object-glass.  To  replace  the  cross-hairs, 
carefully  remove  the  diaphragm  from  the  telescope  tube  and  lay 
it  upon  a  white  surface  with  the  cross-hair  side  upwards.  It  will 
be  noticed  that  there  are  incisions  upon  the  face  of  the  diaphragm 
intended  to  indicate  the  position  of  the  cross-hairs.  Fasten  one 
end  of  the  spider  web  to  the  diaphragm  by  means  of  beeswax  or 
paraffine,  carefully  suspending  the  other  end  of  the  web  over  the 
opposite  incision  upon  the  diaphragm;  after  the  web  has  been 
properly  stretched  by  fastening  it  to  a  match,  fasten  the  web 
down  as  before.  Repeat  the  operation  with  the  other  cross-hair, 
then  replace  the  diaphragm  in  the  instrument,  care  being  taken 
not  to  break  the  hairs.  The  first  adjustment  may  then  be  made. 

The  Dumpy-level.  This  instrument  (see  Fig.  41)  differs 
from  the  Wye-level  in  the  following  points:  the  uprights  which 
carry  the  telescope  are  firmly  attached  to  the  bar,  and  the  level  - 
tube  is  mounted  on  top  of  the  bar.  The  telescope  is  attached 
firmly  to  the  uprights  so  that  the  whole  structure  is  rigid.  At- 
tached to  one  of  the  uprights  is  a  projecting  piece  with  which  one 
end  of  the  level  is  connected  in  such  a  way  as  to  permit  of  a 
slight  horizontal  movement.  The  other  end  of  the  level-tube  is 
fixed  with  two  capstan -headed  nuts  to  permit  of  vertical  adjust- 
ment. 

A  clamp  and  slow- motion  screw  should  be  attached  to  the 
center.  This,  while  not  an  absolute  necessity,  will,  when  clamped, 
prevent  unnecessary  wear  on  the  center  when  the  instrument  is 
carried  upon  the  shoulder.  The  telescope  may  be  either  erecting 
or  inverting,  but  the  latter  is  to  be  preferred. 

The  dumpy-level,  though  not  as  convenient  in  its  adjust- 
ments as  the  Wye-level,  will,  nevertheless,  when  properly  adjust- 
ed, give  equally  good  results;  while  on  account  of  its  simplicity  and 
compactness,  it  is  not  so  liable  to  have  its  adjustments  disturbed. 

Adjustments.  The  adjustments  of  the  dumpy-level  are  two 
in  number  and  should  be  made  in  the  following  order: 

1.  To  make  the  axis  of  the  bubble-tube  perpendicular  to  the 
vertical  axis  of  the  instrument. 


is 

il 

r  as 

l! 

1 


PLANE  SURVEYING 


51 


2.  To  make  the  line  of  collimation  perpendicular  to  the 
vertical  axis  of  the  instrument. 

To  make  the  first  adjustment,  set  up  the  instrument  firmly 
in  a  position  shaded  from  sun  and  wind.  Turn  the  telescope  over 
a  pair  of  opposite  plate  screws,  and  by  means  of  these  screws 
bring  the  bubble  accurately  to  the  center  of  the  tube.  Repeat  the 
operation  over  the  other  pair  of  screws,  and  so  on  alternately  over 
each  pair  of  screws,  until  the  bubble  remains  as  nearly  as  possible 
in  the  center  of  the  tube  for  both  positions  of  the  telescope,  care 
being  taken  not  to  swing  the  telescope  through  more  than  90  de- 
grees. 

Now  turn  the  telescope  accurately  over  a  pair  of  opposite 
plate  screws  and  after  leveling  carefully,  swing  the  telescope 
through  180  degrees  directly  over  the  same  pair  of  screws.  If  the 
bubble  remains  in  the  center  of  the  tube,  the  adjustment  is  corn- 


Fig.  43. 

plete.  If  the  bubble  does  not  remain  in  the  center  of  the  tube, 
bring  it  one-half  way  back  to  the  center  by  means  of  the  vertical 
capstan -headed  screws  at  one  end  of  the  tube,  and  the  remainder 
of  the  distance  by  the  leveling  screws.  Now  repeat  the  test  and 
adjustment  until  the  bubble  remains  in  the  center  of  the  tube 
through  all  positions  of  the  telescope. 

The  second  adjustment  of  the  dumpy-level  must  be  made  by 
the  "Peg  Method".  Select  a  piece  of  ground  as  nearly  level  as 
possible  and  lay  out  a  straight  line  upon  it,  from  400  to  000  feet 
in  length,  driving  a  stake  at  each  end  and  at  the  center.  Set  up 
the  level  over  the  center  stake  and  after  leveling  carefully  direct 
the  telescope  to  the  rod  held  upon  X  (see  Fig.  43),  and  take  the 
reading  by  the  target.  Now  direct  the  telescope  to  the  rod  held 
at  M  and  take  the  reading.  The  difference  of  these  two  rod  read- 
ings at  N  and  M  will  join  the  two  differences  of  elevations,  no 
matter  how  much  the  line  of  collimation  is  out  of  adjustment. 


59 


52  PLANE  SURVEYING 

Now  set  up  the  instrument  within  about  twenty  feet  of  the 
stakes,  as  N,  and  take  the  rod  reading;  carry  the  rod  to  M  and 
take  the  reading.  If  the  difference  of  these  last  two  readings  is 
the  same  as  before,  the  line  of  collimation  is  in  adjustment;  if  not, 
correct  nearly  the  whole  error  by  means  of  the  upper  and  lower 
capstan -headed  screws  attached  to  the  diaphrag  n  carrying  the 
cross-hairs.  Repeat  the  teat  and  correction  severe  I  times  until  the 
difference  of  elevation  from  both  positions  of  the  i  .strument  agree. 

It  may  be  well  to  note  right  here  that  the  se  ond  adjustment 
of  the  Wye-level  may  be  made  by  the  "  Peg  Method,"  but  it  is 
thought  that  the  method  given  is  the  more  convenient. 

The  Precise  Spirit  Level.  The  description  of  this  instrument 
which  is  shown  in  Fig.  44  is  taken  from  the  catalogue  of  the 
makers,  F.  E.  Brandis,  Sons,  &  Co. 

The  principle  underlying  the  construction  of  the  instrument 
is  that  the  telescope  can  be  moved  in  a  vertical  plane  about  a  hor- 
izontal axis  by  means  of  a  micrometer  screw.  This  construction  is 
especially  adapted  to  ^the  object  in  view,  viz:  of  multiplying  the 
pointings  on  a  mark  either  in  the  horizon  of  the  instrument  or  at 
an  angle  above  or  below. 

The  superstructure  consists  of  two  uprights  joined  somewhat 
below  their  middle  by  a  horizontal  plate.  The  upper  portions  of 
the  uprights  are  fashioned  into  Y's,  and  carry  the  telescope  and  the 
striding  level;  the  lower  portions  are  cut  out  so  as  to  leave  guide 
pieces  passing  outside  the  lower  plate.  A  capstan-headed  pivot 
screw  passes  through  each  guide  piece  at  one  end  into  small  sock- 
ets in  the  fixed  plate.  Passing  through  the  fixed  plate,  the  mi- 
crometer screw  moves  between  the  guide  pieces  at  the  other  end 
and  abuts  against  a  small  steel  surface.  The  fixed  plate  carries  an 
index,  and  one  of  the  guide  pieces  a  corresponding  scale  to  register 
the  whole  terms  of  the  micrometer,  and  also  a  pointer  for  reading 
the  subdivisions  of  the  micrometer  head  of  which  there  are  100. 

The  Qurley  Binocular.  The  binocular  hand  level,  as  the 
word  implies,  is  a  hand  level  with  a  double  telescope  attached.  It 
xs  similar  to  the  monocular  hand  level  in  many  respects,  except  it 
is  provided  with  screw  centering  and  focusing  adjustment  and 
can  be  adjusted  to  the  different  widths  of  the  eye,  avoiding  all 
strain  to  the  ocular  muscles. 


60 


PLANE  SURVEYING 


53 


Adjustment.  Follow  the  principles  as  laid  down  concerning 
the  Locke  hand  level. 

The  Qurley  Monocular  Hand  Level.  This  instrument  is  a 
telescope  hand  level,  by  which  readings  are  more  definitely  deter- 
mined on  a  rod  at  some  distance  than  is  possible  with  the  ordinary 
hand  level. 


Adjustment.  Follow  the  principles  as  laid  down  relative  to 
the  Locke  hand  level. 

•«  Setting  up  "  the  Instrument.  The  term  "  setting  up  the 
level"  means  to  place  it  in  position  to  secure  horizontal  sights.  To 
do  this,  plant  the  legs  firmly  in  the  ground  at  approximately 


61 


54 


PLANE  SURVEYING 


equal  distances  apart  so  as  to  make  the  leveling  plate  horizontal. 
Bring  the  telescope  directly  over  and  in  line  with  the  two  leveling 
screws  between  the  plates  and  opposite  each  other.  As  you  stand 
facing  the  instrument  turn  the  thumb  of  the  left  hand  in  the  di- 
rection of  the  motion  of  the  bubble  and  turn  both  thumb  screws 
towards  or  away  from  each  other,  being  careful  not  to  strain  the 
level  plates  by  having  the  leveling  screws  too  tight.  These  screws 
should  bear  firmly  upon  the  plates  and  should  move 
with  ease  ana  smoothness,  but  there  should  be  no  movement 


Fig.  4~->. 


Fig.  46. 


of  the  vertical  axis  of  the  instrument.  Turn  the  screws 
until  the  bubble  appears  along  the  graduations  of  the  bub- 
ble tube  and  bring  it  to  the  middle,  then  turn  the  telescope  at 
right  angles  to  these  two  screws  and  over  the  other  two.  In  like 
manner  perform  the  same  operation  as  before.  This  will  cause  the 
bubble  to  run  away  from  its  former  position;  bring  the  bubble  ac- 
curately in  the  center  of  the  tube  over  these  screws,  that  is,  hav- 
ing equal  spaces  on  each  side  of  the  zero  of  the  scale,  then  turn 
the  telescope  over  the  former  screws  and  bring  the  bubble  in  the 
center.  Do  this  several  times  until  the  bubble  remains  stationary 
at  any  angle  the  telescope  may  turn;  to  test  this,  turn  the  telescope 
half  way  around  and  see  if  the  bubble  moves;  should  it  remain 
stationary,  the  instrument  is  level.  The  level  is,  with  few  excep- 
tions, never  placed  in  line  (except  when  being  adjusted  under  the 


PLANE  SURVEYING  55 


peg  method).  It  is  usually  placed  in  some  convenient  spot  where 
the  greatest  number  of  horizontal  sights  can  be  secured.  As  al- 
ready stated,  the  tripod  legs  must  be  so  placed  as  to  make  the 
plates  horizontal.  This  will  save  time  in  bringing  the  bubble  in 
its  proper  position.  Should  it  be  required  to  set  up  the  instru- 
ment on  the  side  of  a  hill,  place  one  leg  at  an  altitude  and  the  other 
two  in  apparent  line  with  each  other  (see  Figs.  45  and  46), 
but  where  the  tripod  is  adjustable  the  proper  method  is  apparent. 

After  the  instrument  is  set  up  and  leveled,  focus  the  eye- 
piece upon  the  wires  and  focus  the  object-glass  on  the  rod  by 
means  of  the  screw  placed  for  that  purpose  on  the  top  or  side  of 
the  telescope  Care  should  be  taken  not  to  take  a  reading  until 
the  bubble  has  been  carefully  observed  and  brought  in  the  exact 
center  of  the  bubble  tube.  When  this  is  completed,  sight  through 
the  telescope  and  note  the  rod  reading  or  set  the  target  rod;  again 
look  at  the  bubble  and  see  if  it  has  moved  away  from  its  former 
position;  if  not,  again -sight  on  the  rod  and  see  if  the  first  observa- 
tion was  correct.  Should  the  intersection  of  the  cross-hairs  fail 
to  coincide  with  the  horizontal  and  vertical  lines  of  the  target  or 
the  center  of  the  rod,  the  rodman  is  to  incline  the  rod  by  the  sig- 
nals of  the  observer,  until  it  coincides  or  is  in  line  of  collimation. 

Care  of  the  Instrument.  This  duty  properly  belongs  to  the 
instrument  man  or  leveler,  and  the  requirements  should  be  thor- 
oughly understood.  While  in  the  field,  the  instrument  remains 
on  the  tripod  and  is  carried  from  place  to  place  as  the  work  re- 
quires, but  when  taken  any  distance,  such  as  on  railway  trains, 
street  cars,  etc.,  it  should  be  carefully  placed  in  the  box  and  car- 
ried by  one  who  is  capable  of  giving  it  proper  care  and  attention. 

The  instrument  man  being  responsible  for  the  instrument,  it 
is  natural,  and  perhaps  best,  that  he  should  always  carry  the  in- 
strument. In  fact,  the  greatest  amount  of  precaution  should  be 
exercised  in  the  care  of  the  instrument,  both  in  the  field  and  while 
conveying  it. 

Instruments  in  general,  the  level  in  particular,  should  never 
be  unduly  exposed  to  the  rays  of  the  sun,  as  this  will  have  a  tend- 
ency to  throw  its  various  sensitive  parts  out  of  adjustment,  there- 
fore, whenever  possible,  place  the  instrument,  whether  it  is  on  the 
tripod  or  not,  in  the  shade. 


63 


56  PLANE  SURVEYING 

The  leveler  should  always  exercise  great  care  not  to  disturb 
the  instrument  after  it  is  set  up  and  should  avoid,  as  much  as 
possible,  walking  around  it  unreasonably,  especially  if  the  ground 
is  soft,  or  the  position  of  the  instrument  not  very  firm.  This  ap- 
plies to  all  persons  whether  in  the  active  performance  of  duty  or 
not.  It  is  frequently  necessary  to  set  up  the  instrument  in  places 
such  as  loose  timber,  rocks,  etc.,  thus  the  importance  of  this  care  is 
apparent.  If  disturbed  to  any  great  extent  it  will  be  necessary  to 
relevel  it,  and  if  the  position  of  the  legs  of  the  tripod  is  disturbed, 
the  entire  work  must  be  done  over,  because  the  height  of  the  instru- 
ment will  not  be  the  same  as  in  its  former  position.  Should  the  in- 
strument be  disturbed  after  a  turning  point  has  been  established 
and  its  elevation  ascertained,  it  will  only  be  necessary  to  take  a' 
reading  on  the  last  turning  point  to  determine  the  new  height  of 
the  instrument.  After  leveling,  the  instrument  man  should  keep 
his  hands  off  the  instrument  except  for  the  purpose  of  leveling  and 
adjusting  the  telescope.  He  should  not  make  a  practice  of  leaning 
his  weight  on  the  tripod.  It  is  often  necessary  to  send  instruments 
great  distances,  and  in  so  doing,  in  no  case  should  it  be  sent  by 
express  or  freight  without  first  being  properly  packed  and  secured 
against  breakage;  because  of  its-  fine  construction  and  sensitive- 
ness it  may  get  out  of  adjustment  to  such  an  extent  as  to  render 
it  impossible  to  readjust  it  for  good  work  by  any  method  known 
to  the  engineer,  and  may  become  worthless  and  beyond  repair  even 
to  an  instrument  maker. 

The  student  should  appreciate  that  the  care  of  the  instrument 
is  just  as  important  to  good  work  as  its  original  excellence. 

Leveling.  To  determine  the  difference  in  elevation  between 
two  points,  both  of  which  are  visible  from  a  single  position  of  the 
instrument,  set  up  the  instrument  in  such  a  position  that  the  rod 
held  upon  either  point  will  be  visible.  Now  send  the  rod  to  one 
of  the  points  as  at  A  in  Fig.  48;  direct  the  telescope  upon  it  and 
take  the  rod  reading;  now  direct  the  telescope  to  the  rod  held  at 
B  and  again  note  the  reading.  Evidently  the  difference  of  the  rod 
will  give  the  difference  in  elevation  of  the  two  points. 

If  the  points  are  too  far  apart  or  if  the  difference  of  elevation 
is  too  great  to  be  determined  from  one  setting  of  the  instrument, 
intermediate  points  must  be  taken.  For  instance,  suppose  it  is 


64 


PLANE  SURVEYING 


57 


desired  to  find  the  difference  of  elevation  of  A  and  C  in  Fig  47, 
C  being  too  far  below  A  to  permit  of  being  read  upon  both  points 
from  a  single  position  of  the  instrument.  Set  up  the  instrument 
(not  necessarily  on  line  from  A  to  C)  in  some  position  such  that 
the  line  of  sight  will  strike  the  rod  as  near  its  foot  as  it  is  pos- 
sible to  take  a  reading:  send  the  rod  to  some  point  B  such  that 
the  line  of  sight  will  strike  the  rod  near  the  top  when  extended. 
The  difference  of  these  rod  readings  will  give  the  difference  of 
level  of  A  and  B.  Now  carry  the  instrument  to  some  point  such 
that  rod  readings  can  be  taken  upon  B  and  C.  The  difference  of 
the  rod  readings  upon  B  and  C  added  to  the  difference  of  rod  read- 


Fig.  47. 

ings  upon  A  and  B  will  give  the  difference  of  level  of  A  and  C, 
proper  attention  being  given  to  signs. 

If  the  line  of  levels  is  very  extended  the  above  method  is 
awkward,  as  some  of  the  differences  will  be  positive  and 
some  negative.  Choose  some  plane  called  a  datum  plane,  such 
that  all  of  the  points  in  the  line  of  levels  will  lie  above  it. 

Beginning  at  the  point  A,  assume  the  elevation  of  the  point 
above  the  datum  plane.  Read  the  rod  held  upon  A,  and  the  read- 
ing added  to  the  assumed  elevation  will  give  the  height  of  the 
cross-hairs  above  the  datum  plane,  called  the  "height  of  instru- 
ment" (II. I.).  Now,  turn  the  instrument  upon  the  point  B  and 
read  the  rod  and  it  is  evident  that  this  last  rod  reading  subtracted 
from  the  height  of  instrument  will  give  the  elevation  of  B  above 
the  datum  plane.  Next  move  the  instrument  beyond  B,  or  at 
least  where  it  can  command  a  view  of  B  and  C  and  again  sight  to 
the  rod  held  upon  B.  This  last  rod  reading  added  to  the  elevation 
of  B  will  give  the  new  height  of  instrument  from  which  if  the  rod 


65 


58 


PLANE  SURVEYING 


reading  at  C  is  subtracted  will  give  the  elevation  of  C  above  the 
datum  plane.  Fig.  48  will  make  the  method  of  procedure  apparent. 
Eeferring  now  to  that  figure,  the  first  rod  reading  taken  upon 
the  point  A  is  ordinarily  called  a  "back-sight"  and  the  first  read- 
ing taken  upon  B  is  called  a  "fore-sight".  There  seems  to  be  no 
crood  reason  for  adhering  to  this  method  of  distinguishing  between 
the  rod  readings  and  it  is  illogical  and  misleading.  A  back  sight 
is  not  necessarily  taken  behind  the  instrument,  that  is,  in  a  direc- 
tion contrary  to  the  progress  of  the  survey,  neither  is  a  fore-sight 


~5 


\Datum  Plan& 


Fig.  48. 

necessarily  taken  in  front  of  the  instrument.  It  is  more  logical 
and  less  misleading  to  designate  these  rod  readings  by  the  terms 
•-plus-sight"  and  "minus-sight". 

A  plus-sight,  therefore,  is  one  taken  upon  a  point  of  known 
or  assumed  elevation,  to  determine  the  height  of  instrument. 

A  minus-sight  is  one  taken  upon  a  point  of  unknown  elevation 
and  which,  subtracted  from  the  height  of  instrument,  will  give  the 
required  elevation. 

A  "bench-mark"  (B.M.)  is  some  object  of  a  permanent 
character,  the  elevation  of  which,  together  with  its  location,  is 
accurately  determined  for  future  reference  and  for  checking  the 
levels. 

A  "peg",  "plug",  or  turning  point"  (T.P.),  is  a  point  used 
for  the  purpose  of  changing  the  position  of  the  instrument.  This 


PLANE  SURVEYING  59 

turning  point  may  be  taken  upon  a  bench-mark,  but  is  oftener 
taken  upon  the  top  of  a  spike  or  stake  driven  into  the  ground. 

If  a  self-reading  rod  is  used,  the  instrument  man  will  carry 
the  notebook  and  record  the  rod  readings  as  they  are  observed. 
The  leveler  should  cultivate  the  practice  of  calculating  the  ele- 
vations of  his  stations  as  the  work  progresses,  thereby  enabling  him 
to  discern  errors  when  they  occur. 

If  a  target  rod  is  used  upon  the  work,  the  rodman  should  also 
carry  a  notebook  in  which  he  should  at  least  enter  all  readings 
upon  turning  points  and  bench-marks  and  check  up  with  the  in- 
strument man  at  every  opportunity.  Under  the  circumstances, 
the  instrument  man  is  more  or  less  dependent  upon  his  rodman 
for  the  correct  reading  of  the  rod  and  when  an  inexperienced  rod- 
man must  be  employed,  the  self-reading  rod  will  give  the  better 
results. 

The  limit  of  range  of  an  ordinary  leveling  instrument  is  about 
400  feet,  and  sights  should  not  be  taken  at  a  greater  distance. 

The  method  of  keeping  the  field  notes  for  the  work  above  out- 
lined is  given  below.  A  level  notebook  especially  adapted  to 
the  purpose  should  be  procured,  the  notes  entered  on  the  left-hand 
pages,  the  right-hand  pages  being  reserved  for  remarks,  sketches, 
etc. 


STA. 

+  S 

11.  1. 

—  S 

ELEV. 

A 

0.650 

1000.650 

1000.00 

©  N.    E.    cor. 

B 

1.250 

993.140 

8.760 

991.890 

of  abutment 

0 

2.380 

987.670 

7.850 

985.290 

Main    street 

D 

9.570 

978.100 

bridge. 

It  will  be  noticed  that  the  algebraic  sum  of  the  plus  and 
minus  readings  equals  the  difference  of  elevation  of  the  first  and 
last  stations,  and  these  quantities  should  be  checked  as  often  as 
possible  to  discover  errors  in  addition  or  subtraction. 

Profile  Leveling.  The  method  of  profile  leveling  is  the  same 
in  principle  as  above  outlined,  but  the  details  of  field  work  are  a 
little  different. 

In  this  sort  of  work  it  is  intended  to  determine  a  vertical 
section  of  the  ground  above  a  datum  plane.  To  this  end,  rod  read- 
ings are  taken  sufficiently  close  together  that  when  the  elevations 


67 


60 


PLANE  SURVEYING 


are  plotted  and  the  points  connected,  the  resulting  irregular  line 
will  closely  approximate  the  actual  line  of  the  surface. 

Profile  levels  are  usually  run  in  connection  with  a  transit  or 
chain  survey  of  the  line,  the  positions  of  the  points  being  first 
established  upon  railroad  surveys.  These  points  are  usually  100 
feet  apart  unless  the  ground  is  very  irregular,  when  they  may  be 
50  or  25  feet  apart  or  even  less,  the  points  being  indicated  by 
stakes.  Upon  sewer  or  street  work  they  should  seldom  be  more 
than  50  feet  apart  and  the  readings  should  be  taken  with  the  rod 
held  upon  the  ground. 

Fig.  49  will  illustrate  the  difference  between  profile  leveling 
and  the  first  system  outlined,  sometimes  called  differential  leveling 
or  "peg"  leveling.  Referring  now  to  that  figure  A  Bis  the  datum 


Fig.  49. 

plane  and  the  full  lines  at  C,D,  and  E  represent  positions  of  the 
rod  for  turning  points.  Assuming  the  elevation  of  the  point  C, 
the  rod  is  held  upon  it  and  the  reading  added  to  the  elevation  for 
the  height  of  instrument.  The  rod  is  then  carried  successively  to 
the  points  a,  J,  c,  <7,  and  each  reading  is  in  turn  subtracted  from 
the  height  of  instrument  at  C,  to  get  the  elevations  of  these  points. 

The  rod  is  then  held  upon  the  point  D,  the  instrument  moved 
and  the  plus-sight  upon  D  added  to  the  elevation  for  the  new 
height  of  instrument.  The  rod  readings  upon  <?,  f,  y,  h,  /,  etc., 
are  then  each  subtracted  from  this  new  height  of  instrument  for 
elevations. 

This  figure  illustrates  the  improper  use  of  th^  terms  back- 
sight and  .fore-sight.  The  rod  readings  at  C,  </,  and  1>  are  taken 


68 


PLANK  SURVEYING 


01 


behind    the    instrument,    but    the  rod    reading  at  C  is  the  only 
plus-sight. 

The  method  of  keeping  the  field  notes  is  illustrated  below. 


STA. 


O. 

-f  50 
1 

+  50 
T.  P.  +  62 

2 

+  50 
3 

+  50 
4 


+  S 
3.25 


2.64 


H.I. 

585.70 


581.49 


—  S 

3.78 

4.18 
5.06 
6.85 
3.10 
3.18 
3.90 
4.60 


ELEV. 
582.45 
581.92 
581.52 
580.64 
578.85 
578.39 
578.31 
577.59 
576.89 
576.24 


CROSS=SECTIONING. 

One  of  the  most  important  problems  that  confronts  the 
leveler  is  the  setting  of  "slope  stakes,"  called  cross-sectioning, 
from. which  may  be  determined  the  amount  of  earthwork  in  cut 
or  fill,  and  which  mark  the  extreme  limits  of  the  operations  of  the 
construction  corps  in  building  railways,  highways,  sewers,  canals, 
irrigation  ditches,  etc. 

The  problem  is  as  follows:  Given  the  required  width  of 
finished  roadbed  or  channel,  with  proper  side  slopes  (depending 


upon  the  kind  of  material),  it  is  required  to  determine  where 
these  side  slopes  will  intersect  the  natural  surface  of  the  ground 
with  reference  to  the  center  line  of  the  survey.  The  center  line  is 
defined  by  stake,  carefully  aligned  and  leveled,  and  a  profile  of  it 
is  prepared  upon  which  the  grade  line  is  laid  down,  showing  the 


62 


PLANE  SURVEYING 


elevation  of  the  finished  roadbed  or  channel  with  reference  to  the 
natural  surface  of  the  ground. 

Let  us  assume  the  ground  to  be  level  transverse  to  the  center 
line.  Depth  of  cut  at  center  =  12  feet;  side  slopes  1|  feet  hori- 
zontal to  1  foot  vertical;  width  of  cut  at  bottom  —  20  feet.  See 
Fig.  50. 

Set  up  the  instrument  in  some  convenient  position  that  will 
command  a  view  of  as  much  ground  as  possible.  Hold  the  rod 
upon  the  ground  at  the  center  stake  and  note  the  reading.  Sup- 
pose it  to  be  3.5  feet.  Now  if  the  ground  is  level,  the  distance 
from  C  to  B  is  evidently  10  +  (12xH)  =28  feet  and  the  rod 
should  again  read  3.5  feet  when  held  at  B.  The  point  A  would 
be  found  in  the  same  way. 

The  notes  would  be  kept  as  shown  below. 


Sta. 

Dis. 

Left 

Center 

Right' 

Area 

C.Yds. 

175 
170 
176 

50 
50 

+  12.0 
28~ 
+   3.0 
14~5 
+    2.5 

+  12.0 
+  6.0 
+  5.0 

+  7 
9 

+  12.0 

28 
+  9.2 
2OO 
+   8.0 
22~ 

13.751 

The  preceding  example  illustrates  one  of  the  simplest  cases 
that  occur  in  practice.  Let  us  now  take  the  case  of  a  line  located 
upon  the  side  of  a  hill.  See  Fig.  51. 

Depth  to  grade  at  center  6  feet;  width  at  bottom  20  feet; 
side  slopes  1|  to  1.  As  before,  hold  the  rod  upon  the  ground  at 
C  and  determine  the  height  of  instrument  abov.e  C.  Suppose  this 
to  be  5.5  feet.  Now,  if  the  ground  were  level  through  C  it  would 
be  necessary  to  measure  to  the  right  10  +  (0  X  1|)  =  19  feet  to  the 
point  D  and  the  rod  should  read  5.5  feet.  Instead  it  reads,  say 
2.8  feet.  We  know  therefore  that  we  have  not  gone  out  far 
enough  by  (5.5 — 2.8)  H  —  4.05  feet,  if  the  ground  were  level 
through  the  point  D,  bringing  us  to  the  point  E  where  the  rod 
should  read  2.8  feet.  Suppose  it  reads  2.3  feet.  We  must  then 
go  out  0.75  foot  farther,  each  move  bringing  us  closer  and  closer 
to  the  point  B.  This  operation  may  be  repeated  as  often  as  is 
considered  necessary,  but  with  a  little  experience  in  this  sort  of 


70 


PLANE  SURVEYING 


63 


work  the  instrument  man  can  direct  the  rod  closely  enough  to  the 
point  B  for  all  practical  purposes.  We  then  enter  the  notes  in 
the  second  line  of  the  record  shown  above. 

Upon  the  left  of  the  center,  these  operations  are  reversed. 
That  is  to  say,  we  measure  out  19  feet  and  instead  of  the  rod  read- 
ing 5.5  feet,  it  reads,  say,  8.5  feet.  We  know  then  that  we  are 
out  too  far  by  4.5  feet.  We  then  move  in  toward  the  center  the 


Fig.  51. 

required  distance  and  read  the  rod  again,  noting  how  much  it  dif- 
fers from  8.5  feet,  if  any,  and  enter  the  final  results  in  the  notes. 

A  third  case  is  shown  in  Fig.  52,  in  which  the  transverse 
slope  is  not  uniform.  The  method  of  procedure  is  the  same  as  in 
the  other  cases,  but  the  rod  should  be  held  at  the  point  where  the 
slope  changes  in  order  to  find  its  height  above  grade.  Enter  this 
and  the  distance  out  in  the  third  line  of  the  notes. 

The  transverse  section  may  be  very  irregular,  in  which  case 
it  may  be  necessary  to  take  readings  at  several  points  in  order  to 


Pig.  52. 

calculate  the  area  of  the  sections  with  more  exactness.     At  times 
a  section  will  be  cut  partly  in  rock  and  partly  in  earth,  forming  a 


71 


64 


PLANE  SURVEYING 


compound  section.  Each  material  will,  of  course,  have  its  own 
proper  side  slope,  and  the  depth  and  extent  of  each  must  be  deter- 
mined by  soundings. 

In  case  the  section  is  in  fill  instead  of  in  cut,  the  method  is 
the  same  as  in  the  preceding  cases,  as  will  be  illustrated  in  the  fol- 
lowing examples.  Let  us  first  take  a  section  level  transversely. 

See  Fig.  53. 

In  this  case  the  finished  grade  is  to  be  9  feet  above  the  point 


Fig.  53. 

C.  Hold  the  rod  at  C  and  suppose  it  reads  3.25  feet.  Now  since 
the  ground  is  level  we  go  out  to  the  right  and  left  9-|-(9xl4) 
=  22.5  feet  and  set  the  stakes  at  A  and  B  entering  the  record  in 
the  notebook  as  before,  except  that  now  the  numerator  of  the 
fraction  will  be  marked  — instead  of  +  . 

We  will  next  take  the  case  where  the  surface  of  the  ground 
has  a  transverse  slope.  See  Fig.  54.  Now  hold  the  rod  at  the 
point  C,  and  suppose  it  reads  9.25  feet.  Now  if  the  ground  were 


i« is'-  1 svajy- 


Fig.  54. 

level  through  C  we  would  have  to  go  out  to  the  right  9 +(6. 25 
Xl.5)  =  18.4  feet  to  some  point  D.  But  there  the  rod  reads,  say, 
1.5  feet,  hence  we  know  we  are  out  too  far  by  7.75x1.5  =  11.63 
feet,  bringing  us  back  to  some  point  as  E  and  the  rod  now  reads, 


PLANE  SURVEYING  65 


say,  3.5  feet  and  we  move  out  again  2.0  X  1.5  =  3  feet.  Therefore 
we  move  back  and  forth  until  we  find  the  point  B  where  the  com- 
puted rod  reading  and  the  actual  reading  agree. 

Sometimes  it  will  be  found  that  a  part  of  the  section  is  in 
cut  and  a  part  in  fill,  but  methods  outlined  will  serve  in  any  case. 

The  distance  between  the  sections  longitudinally  will  depend 
upon  the  nature  of  the  ground.  On  uniformly  sloping  or  level 
ground  they  may  be  taken  100  feet  apart.  Over  uneven  ground 
it  may  be  necessary  to  take  them  as  closely  together  as  25  feet  or 
even  less.  In  the  sections  themselves,  a  sufficient  number  of  rod 
readings  should  be  taken  that  the  area  of  the  sections  may  be 
determined  with  reasonable  accuracy. 

After  the  field  work  is  completed,  the  notes  are  plotted, 
usually  upon  cross-section  paper,  and  the  areas  determined  either 
with  a  planimeter,  by  Simpson's  rule  or  some  other  method.  These 
sections  then  divide  the  earthwork  into  a  system  of  prismoids  of 
which  the  volume  must  be  calculated.  The  formula  for  calculat- 
ing volumes  is  known  as  the  Prismoidal  Formula  and  is  as  follows: 


in  which  I  =  length  between  consecutive  sections,  A  =  one  end  sec- 
tion, B  =  the  other  end  section  and  M  =  the  section  midway  be- 
tween the  two.  The  result  is  given  in  cubic  yards. 

The  mistake  must  not  be  made  of  assuming  that  M  is  a  mean 
between  A  and  B;  but  a  theoretical  section  must  be  plotted  whose 
dimensions  are  a  mean  between  those  of  A  and  B.  This  often 
results  in  quite  a  complicated  problem,  and  various  other  formulas 
have  been  devised  to  give  sufficiently  close  results  without  the 
labor  and  time  involved  in  the  preceding.  This  will  be  treated  in 
detail  in  Railroad  Engineering. 


73 


COAST  SURVEY  PARTY  STARTING  TO  WORK  FROM  A  TRIANGULATION  STATION 

The  topographer  and  his  assistant  are  adjusting  the  instrument  under  the  "signal;"  the 
instrument  man  is  at  his  never-ending  task  of  putting  needle-points  on  pencils:  one  of  the 
rodmen  (with  telemeter  rod)  Is  waiting  for  instructions  to  set  out,  and  the  other  is  pit-king 
up  other  "signals"  with  the  glasses. 


PLANE   SURVEYING. 

PART  II. 


The  meridian  plane  of  any  place  upon  the  earth's  surface 
is  a  great  circle  passing  through  the  zenith  of  the  place  and  the 
poles  of  the  earth.  A  true  meridian  is,  therefore,  a  line  lying  in 
this  plane,  and  would,  if  produced,  pass  through  the  poles. 

The  magnetic  meridian  plane  would  in  the  same  way  be 
defined  by  the  zenith  and  the  magnetic  pole  of  the  earth ;  but  since 
this  pole  is  not  fixed  in  position,  the  magnetic  meridian  is  defined 
as  the  direction  of  the  line  indicated  by  the  position  of  the 
magnetic  needle. 

At  a  few  places  upon  the  surface  of  the  earth,  the  true  meridian 
and  the  magnetic  meridian  coincide  at  times,  but  for  the  most 
part  they  differ  in  direction  by  an  ever  varying  quantity.  The 
angle  at  any  place  between  the  true  meridian  and  the  meridian  as 
defined  by  the  magnetic  needle,  is  called  the  magnetic  declination 
for  that  place.  If  the  direction  of  the  magnetic  meridian  were 
constant,  or  if  the  changes  followed  any  particular  law,  it  would 
be  a  comparatively  simple  matter  to  determine  the  declination  for 
any  time  or  place.  The  variations  occurring  are  of  three  principal 
kinds — diurnal,  annual,  and  secular,  the  last  being  the  most 
important. 

Diurnal  Variation.  On  continuing  observations  of  the 
direction  of  the  needle  throughout  the  day,  it  will  be  found  that 
the  north  end  of  the  needle  will  move  in  one  direction  from  about 
8  A.  M.  until  shortly  after  noon,  and  then  gradually  return  to  its 
former  position. 

Annual  Variation.  If  observations  be  continued  throughout 
the  year,  it  will  be  found  that  the  diurnal  changes  vary  with  the 
seasons,  being  greater  in  summer  than  in  winter. 

Secular  Variation.  If  accurate  observations  on  the  declination 
of  the  needle,  in  the  same  place,  are  continued  over  a  number  of 
years,  it  will  be  found  that  there  is  a  continual  and  comparatively 
constant  increase  or  decrease  of  the  declination,  continuing  in  the 
same  direction  over  a  long  period  of  years. 

Copyright,  19O8,  by  American  School  of  Correspondence. 


68  PLANE   SURVEYING 

Besides  the  above,  the  declination  is  subject  to  variations  more 
or  less  irregular,  due  to  local  conditions,  lunar  perturbations,  sun 
spots,  magnetic  storms,  etc. 

The  declination  in  any  part  of  the  United  States  may  be 
approximately  determined  by  consulting  the  chart  issued  from  time 
to  time  by  the  United  States  Coast  and  Geodetic  Survey.  (See 
chart,  page  132.)  Upon  this  chart  all  points  at  which  the  needle 
points  to  the  true  north  are  connected  by  lines,  called  agonic  lines 
or  lines  of  no  declination.  Lines  are  also  drawn  connecting  points 
of  the  same  declination,  called  isogonic  lines. 

The  isogonic  curves  or  lines  of  equal  magnetic  declination  (variation 
of  compass)  are  drawn  for  each  degree,  a  +  sign  indicating  West  declination, 
a  —  sign  indicating  East  declination. 

The  magnetic  needle  will  point  due  North  at  all  places  through  which 
the  agonic  or  zero  line  passes,  as  indicated  on  the  chart. 

Before  undertaking  an  extensive  or  important  survey,  it  is 
the  first  duty  of  the  surveyor  to  determine  accurately  his  declina- 
tion. This  is  best  done  by  laying  out  a  true  meridian  upon  the 
ground  and  comparing  its  direction  with  that  indicated  by  the 
needle.  Before  describing  the  methods  of  laying  out  a  true 
meridian,  it  will  be  best  to  describe  the  compass. 

THE  COMPASS. 

Construction.  The  surveyor's  compass  consists  primarily  of 
a  circular  brass  box,  carrying,  upon  a  pivot  in  its  center,  a  strongly 
magnetized  needle  (see  Fig.  56).  The  inside  edge  of  the  box  on  a 
level  with  the  needle,  is  usually  graduated  to  half  degrees,  and 
smaller  intervals  may  be  "  estimated."  Two  points  diametrically 
opposite  each  other  are  marked  0°,  and  form  the  north  and  south 
ends  of  the  box,  the  south  end  being  indicated  by  the  letter  S,  and 
the  north  end  either  by  the  letter  N  or  by  a  fleur-de-lis  or  other 
striking  figure.  The  divisions  extend  through  90°  upon  both  sides 
of  these  points,  to  the  east  and  west  points  marked  respectively 
E  and  W.  The  east  side  of  the  box,  however,  is  on  the  left' as  the 
observer  faces  the  north  end;  this  is  because  the  needle  remains 
stationary  while  the  box  revolves  around  it.  The  divided  circle  is 
sometimes  movable,  being  fitted  with  a  clamp  and  tangent-screw 
for  setting  off  the  declination  of  the  needle. 


76 


PLANE   SURVEYING 


The  magnetic  needle  is  the  most  essential  part  of  the  compass. 
It  consists  of  a  slender  bar  of  steel,  usually  five  or  six  inches  long, 
strongly  magnetized,  and  balanced  on  a  pivot  so  that  it  may  turn 
freely  and  thus  continue  to  point  in  the  same  direction  however 
much  the  box  carrying  the  pivot  may  be  tunied  around.  To  this 
end  the  pivot  should  be  of  the  hardest  steel,  ground  to  a  very 
fine  point,  or,  better  still,  of  iridium;  and  the  center  of  the  needle 
resting  upon  the  pivot  should  be  fitted  with  a  cap  of  agate  or  other 
hard  substance. 

To  distinguish  the  ends  of  the  needle,  the  north  end  is  usually 


Fig.  56. 

cut  into  a  more  ornamental  form  than  the  south  end,  or  the  latter 
end  may  be  recognized  by  its  carrying  a  coil  of  wire  to  balance  the 
"dip." 

Intensity  of  directive  force  and  sensitiveness  are  the  chief 
requisites  in  a  magnetic  needle,  and  nothing  is  gained  by  making 
a  needle  over  five  inches  in  length.  Indeed,  longer  needles  are 
liable  to  have  their  magnetic  properties  impaired  by  polarization. 
The  needle  should  not  come  to  rest  too  quickly.  Its  sensitiveness 
is  indicated  by  the  number  of  vibrations  that  it  makes  in  a  small 
space  before  coming  to  rest.  Should  it  come  to  rest  quickly  or  be 
sluggish  in  movement,  it  indicates  either  that  the  magnetization 


77 


PLANE   SURVEYING 


is  weak  or  that  there  is  undue  friction  between  needle  and  pivot. 
The  under  side  of  the  box  should  be  fitted  with  a  screw  which, 
engaging  a  lever  upon  the  inside  of  the  box,  will  serve  to  lift  the 
needle  off  the  pivot  when  the  instrument  is  carried  about. 

The  sights  form  the  next  most  important  feature  of  the 
compass.  They  consist  of  two  brass  uprights,  with  a  narrow  slit 
in  each,  terminated  at  intervals  by  circular  apertures.  They  are 
mounted  directly  upon  the  compass-box ;  or  the  bottom  of  the  box 
may  be  extended  at  each  end  in  the  form  of  a  plate,  and  the  sights 
attached  at  the  ends  of  the  plates.  How- 
ever mounted,  the  sights  should  have 
their  slits  in  line  directly  over  the  north 
and  south  points  of  the  divided  circle. 
The  right  and  left  edges,  respectively,  of 
the  sights,  may  have  an  eye-piece  and  a 
series  of  graduations,  by  which  angles 
of  elevation  and  depression  for  a  range 
of  about  twenty  degrees  each  way  can 
be  taken  with  considerable  accuracy. 
This  device  is  called  a  tangent  scale, 
so  called  because  of  the  distance  of  the 
engraved  lines  from  the  O°  line  being 
tangents  (with  a  radius  equal  to  the 
Pig.  57.  distance  between  the  sights)  of  the 

angles  corresponding  to  the  numbers  of  the  lines. 

The  spirit  levels  may  be  placed  at  right  angles  to  each  other 
in  the  bottom  of  the  compass-box,  or  mounted  in  the  same  way 
upon  the  plate. 

The  compass  is  usually  fitted  to  a  spindle  made  slightly 
conical,  which  has  on  its  lower  end  a  ball  turned  perfectly  spherical, 
confined  in  a  socket  by  a  pressure  so  light  that  the  ball  can  be 
moved  in  any  direction  in  leveling  the  instrument.  The  ball  is 
placed  either  in  the  brass  head  of  a  Jacob  staff,  or,  better,  in  the 
top  casting  of  a  tripod. 

A  plumb-bob  should  be  provided  with  the  instrument  to  center 
it  over  a  stake. 

A  telescope  is  sometimes  provided,  to  be  attached  to  one  of 
the  vertical  sights,  for  the  purpose  of  more  clearly  defining  the 


78 


PLANE   SURVEYING  71 

line  of  sight,  The  compass  is,  however,  so  inaccurate  that  it  would 
seem  to  be  an  unnecessary  refinement. 

Prismatic  Compass.  This  is  a  form  of  compass  used  in 
general  where  merely  ordinary  work  is  required.  It  is  about 
3  inches  in  diameter  with  a  floating  metal  dial  (see  Fig.  57),  and  is 
provided  with  folding  sights  and  prism.  By  means  of  the  latter  it 
may  be  read  while  being  pointed.  This  is  especially  useful  when 
the  instrument  is  held  in  the  hand.  Although  it  can  be  mounted 
on  a  Jacob  staff,  it  is  usually  held  in  the  hand  and  carried  in  the 
observer's  pocket. 

Adjustment.  To  Adjust  the  Levels.  First  bring  the  bubbles 
to  the  middle  of  the  tube  by  the  pressure  of  the  hand  on  different 
parts  of  the  plate,  and  then  turn  the  instrument  half-way  round. 
If  the  bubbles  remain  in  the  middle  of  the  tubes,  the  tubes  are  in 
adjustment.  If  the  bubbles  do  not  remain  in  the  middle,  raise 
or  lower  one  end  of  the  tube  to  correct  one-half  the  error.  Relevel 
the  instrument,  again  test,  and  apply  the  correction  as  before. 
Continue  the  operation  until  the  levels  are  in  perfect  adjustment. 

To  Adjust  the  Needle  to  the  "J9«>."  While  the  compass 
is  still  in  a  perfectly  level  condition,  see  if  the  needle  is  in  a 
horizontal  plane.  Should  this  not  be  the  case,  move  the  small  coil 
of  wire  towards  the  high  end  until  the  needle  swings  horizontally. 

To  Adjust  the  Sight- Vanes.  Observe  through  the  slits  a 
fine  hair  or  thread  made  exactly  vertical  by  a  plummet.  Should 
the  hair  appear  on  the  side  of  the  slit,  the  sight-vane  must  be 
adjusted  by  filing  its  under  surface  on  the  side  that  seems  the 
higher. 

To  Adjust  the  Needle.  Having  the  eye  nearly  in  the  same 
plane  with  the  graduated  rim  of  the  compass-box,  bring  one 
end  of  the  needle  in  line  with  any  prominent  graduation  mark  in 
the  circle,  as,  for  instance,  the  zero  or  the  90-degree  mark,  and 
notice  if  the  other  end  corresponds  with  the  same  degree  upon  the 
opposite  side;  if  it  does,  the  needle  is  said  to  "cut"  opposite 
degrees;  if  not,  bend  the  center  pin,  until  the  ends  of  the  needle 
are  brought  into  line  with  the  opposite  degrees. 

Then,  holding  the  needle  in  the  same  position,  turn  the 
instrument  half-way  round,  and  note  whether  the  needle  now  cuts 
opposite  degrees;  if  not,  correct  one-half  the  error  by  bending  the 


79 


PLANE   SURVEYING 


needle,  and  the  other  half  by  bending  the  center  pin.  The 
operation  of  testing  and  correcting  should  be  repeated  until  perfect 
reversion  is  secured  in  the  first  position.  This  being  obtained,  the 
operation  should  be  tried  on  another  quarter  of  the  circle;  if  any 
error  is  found,  the  correction  must  be  made  in  the  center  pin  only, 
the  needle  being  already  straightened  by  the  previous  operation. 
When  the  needle  is  again  made  to  cut,  the  test  should  be  tried  in 
the  other  quarters  of  the  circle,  and  the  correction  made  in  the  same 
manner,  until  the  error  is  entirely  removed  and  the  needle  will 
reverse  at  every  point  of  the  graduated  circle. 

Use.  In  the  operation  of  locating  points,  and  therefore  lines, 
by  angle-measuring  instruments,  two  operations  are  necessary:— 
(1)  to  measure  the  angle  at  the  instrument  between  some  given  line 
and  the  line  passing  through  the  given  point;  (2)  to  measure  the 
distance  from  the  instrument  to  the  given  point.  For  the  first 
operation  two  types  of  instrument  are  in  general  use  —  the 
compass  and  the  transit.  For  the  compass,  the  line  of  reference 
from  which  all  angles  are  measured  is  a  meridian,  and  the  angular 
deviation  from  this  line  is  called  the  bearing.  The  bearing  and 
length  of  a  line  are  collectively  named  the  course.  The  compass, 
therefore,  measures  bearings  directly  and  angles  indirectly. 

To  Determine  the  Bearing  of  One  Point  from  Another. 
"Set  up"  the  compass  over  one  of  the  points,  and  level  carefully. 
Turn  the  sight-vanes  in  the  direction  of  the  second  point,  with  the 
north  end  of  the  plate  ahead.  Hold  a  rod  upon  the  second  point, 
and  cover  it  with  the  slits  in  the  sight-vanes.  Now  lower  the 
needle  upon  the  pivot,  being  sure  that  the  instrument  is  still  level; 
allow  it  to  come  to  rest,  and  read  the  bearing. 

To  Survey  a  Series  of  Lines  with  the  Compass.  "Set  up" 
the  compass  over  the  point  A,  with  the  north  end  of  the  plate 
ahead  (Fig.  58);  and  after  leveling,  turn  the  sight-vanes  to  cover 
a  rod  held  upon  the  point  B.  Now  send  out  the  tape  in  the  direction 
of  B,  and,  sighting  through  the  slits,  signal  the  head  tapeman 
into  line.  Continue  this  until  the  point  B  is  reached.  Now  read 
and  record  the  bearing  and  the  length  of  the  line.  Take  up  the 
instrument,  and  carry  it  to  B.  Set  it  up  over  B,  with  the  north  end 
ahead,  that  is,  pointing  in  the  direction  of  the  survey.  Level,  and 
turn  the  south  end  so  as  to  cover  a  rod  held  upon  the  point  A. 


PLANE   SURVEYING  73 

Read  the  bearing  as  a  check  upon  the  former  one,  but  reversed  in 
direction;  i.  e,,  if  the  bearing  from  A  to  B  was  north  by  east,  the 
bearing  from  B  to  A  will  be  south  by  west.  If  the  direct  and 
reversed  bearings  check,  turn  the  north  end  of  the  compass  to  cover 
a  rod  held  upon  C.  Read  the  bearing,  measure  B  C,  take  the 
instrument  to  C,  and  proceed  as  before. 

If  at  any  station,  such  as  C,  the  direct  and  reversed  bearings 
do  not  agree,  take  the  instrument  back  to  B  and  again  take  the 
bearing  of  B  C.  If  they  still  disagree;  it  indicates  local  attraction 
at  C.  Take  the  instrument  to  D  and  take  the  bearing  of  D  C, 


Fig.  58. 

comparing  it  with  the  bearing  of  C  D.  If  these  disagree,  record 
the  bearings  of  B  C  and  D  C  as  well  as  those  of  C  B  and  C  D. 
The  latter  should  check  the  former,  since  the  local  attraction  at  C 
will  affect  both  lines  equally;  and  the  correct  angle  between  the 
lines  can  be  computed. 

Locating  a  series  of  lines  with  certain  lengths  and  bearings  is 
essentially  the  same  as  above,  except  that  after  the  compass  has 
been  turned  in  the  proper  direction,  the  stations  must  be  brought 
into  proper  line. 

Here  it  may  be  well  to  remark  upon  the  proper  method  of 
reading  and  recording  bearings.  Always  read  the  north  or  south 
end  of  the  plate  first;  i.  e.,  if  a  line  has  a  bearing  35°  east  of 
north,  it  should  be  read  and  recorded  N  35°  E.  If  the  bearing  is 
90°  east  or  west  of  north  or  south,  record  the  bearing  as  E  or  W. 

The  Gunter's  chain  is  always  used  in  land  surveys  made  with 
the  compass,  and  deeds  and  records  of  such  surveys  are  based 
upon  the  Gunter's  chain  as  the  unit. 

Hints  Regarding  the  Use  of  the  Compass.  Sometimes,  as 
when  the  line  of  which  the  bearing  is  required  consists  of  a  fence, 


74  £LANE   SURVEYING 

etc.,  the  compass  cannot  be  set  upon  the  line.  In  such  a  case 
measure  off  equal  distances  at  right  angles  to  the  line,  and  find  the 
bearing  of  the  parallel  line;  the  length  should  be  measured  upon 
the  line  itself.  In  other  cases  it  may  be  more  convenient  to  set 
the  compass  or  rod  "in  line"  upon  the  line  produced,  or  upon  some 
intermediate  point  of  the  line. 

It  is  more  important  to  have  the  compass  level,  crosswise  of 
the  sights,  than  parallel  with  them. 

Avoid  reading  the  bearing  from  the  wrong  number  of  the  two 
between  which  the  needle  points,  as  for  instance  35°  for  25°. 

Check  the  vibrations  of  the  needle  by  gently  raising  it  off  the 
pivot  and  lowering  it  again  by  means  of  the  screw  on  the  under 
side  of  the  box. 

If  the  needle  is  slow  in  starting,  smartly  tap  the  compass  to 
destroy  the  effect  of  any  possible  adhesion  to  the  pivot  or  friction 
of  dust  upon  it. 

Avoid  holding  the  pins,  axe,  or  any  other  body  of  iron,  in 
close  proximity  to  the  needle. 

Should  the  needle  adhere  to  the  glass  after  the  latter  has  been 
dusted  with  a  handkerchief  or  has  been  carried  so  as  to  rub 
against  the  clothes,  the  trouble  is  due  to  the  glass  being  thereby 
charged  with  electricity  and  may  be  obviated  by  moistening  the 
finger  and  applying  it  to  the~glass. 

RELOCATION. 

Suppose  it  is  required  to  relocate  a  line,  no  trace  of  the  old 
survey  being  at  hand  except  the  given  line.  Now,  between  the 
date  of  the  old  survey  and  the  present,  the  declination  of  the  needle 
has  changed  several  degrees.  The  first  duty  of  the  surveyor  is  to 
consider  this  question  very  carefully,  and  to  ascertain  the  probable 
amount  of  change  in  the  magnetic  needle.  Suppose  the  result  of 
his  inquiry  leads  to  N  38°  15'  E  as  the  bearing.  Starting  at 
corner  A,  Fig.  59,  the  surveyor  runs  a  random  line  AS  on  the 
bearing  N  38°  15'  E,  and  measures  along  this  line  a  distance  of 
32  chains,  or  2,112  feet,  to  point  S.  On  arriving  at  S,  the  surveyor 
proceeds  to  look  over  the  ground  on  both  sides  of  this  point  for  a 
lost  corner,  which  is  described  in  the  old  record  as  a  monument, 
stump,  or  some  other  well-defined  mark.  If,  after  diligent  search, 


PLANE   SUKVEYINQ 


no  trace  of  this  mark  can  be  found,  nothing  further  can  be  done 
from  the  data  at  hand.     However,  should  the  mark  be  found  at  m, 
a  perpendicular  is  dropped  upon  the  line  AS,  arid  its  length  is 
measured,  as  is  also  the  distance  nS. 
It  is  now  evident  that   the  distance 
An  becomes  known.     From  the  right 
triangle,  the  angle  nAm  can  be  com- 
puted, and  the  present  magnetic  bear- 
Fig.  59.  ing  of  Am  can  be  determined. 
For  example,  suppose  that  mn  is  found  to  be  37.4  feet,  while 

An  is  2,110.5  feet,  then  tan.  nAm=-^-  —  0.01772,  whence  nAm 

An 

=  1°  01',  and  the  present  magnetic  bearing  of  Am  is  X  39°  16'  E- 
The  distance  Am  =  ^f^^— =  2,110.84  feet.  This  indicates 

that  the  present  work  is  correct,  and  that  the  old  survey  was  in  error 
by  1.16  feet.  As  there  is  a  principle  of  law  that  establishes  corners 
and  monuments,  resurveys  must  control;  therefore  the  new  record 
of  the  line  Am  is  N  39°  1C'  E,  2,110.84  feet.  Intermediate  points 
of  the  line  Am  may  now  be  established  from  the  starting  point  A, 
running  it  out  with  the  new  bearing. 

EXAMPLE   FOR    PRACTICE. 

Compute  the  distance  and  bearing  of  two  points  which  are  not 
intervisible.  Call  the  line  GH.  A  line  is  run  approximately  near 
H,  from  the  known  corner  G  to  a  point  A  which  is  visible  to  H; 
the  bearing  and  length  of  this  line  being  N  42°  15'  E,  714.5  feet: 
AH  being  N  1°  08'  E,  210.5  feet. 

Ans.  N  33°  14'  E,  883.24  feet. 

To  Find  the  Bearing  of  One  Line  to  Another.  Suppose,  in 
Fig.  60,  that  of  the  tract  of  land  therein  described  there  has  been 
prepared  a  rough  plot  upon  which  the  angles,  bearings,  and 
distances  as  taken  from  the  field  book  are  figured.  In  order  to  find 
the  bearing  of  one  line  to  another,  add  together  the  interior  angles 
formed  at  all  the  corners;  call  their  sums  a;  multiply  the  number 
of  the  sides  by  180°;  from  the  product  subtract  360°.  If  the 
remainder  is  equal  to  «,  this  is  proof  that  the  angles  -have  been 
accurately  measured,  This,  however,  will  rarely  if  ever  occur; 


83 


76  PLANE   SURVEYING 

there  will  always  be  some  discrepancy,  but  if  the  field  work  has 
been  performed  with  reasonable  care  the  discrepancy  will  not 
exceed  two  minutes  for  each  angle.  In  this  case  divide  it,  in  equal 
parts,  among  all  the  angles,  adding  or  subtracting,  as  the  case  may 
be,  until  it  amounts  to  less  than  one  minute  for  .each  angle,  when 
it  may  be  entirely  disregarded  in  common  farm  surveys. 

The  corrected  angles  may  now  be  marked  on  the  plot  in  ink, 
and  the  penciled  figures  erased.     We  shall  suppose  the  corrected 


Fig.  60. 

ones  to  be  as  shown  in  Fig.  60.  Next,  by  means  of  these  corrected 
angles,  correct  the  bearings  also. 

Select  some  side,  the  longer  the  better,  from  two  ends  of  which 
the  bearing  and  the  reverse  bearing  agree,  thus  showing  that  the 
bearing  was  probably  not  influenced  by  local  attraction.  Let  side 
2  be  the  one  so  selected;  assume  its  bearing  N  75°  32'  E,  as  taken 
on  the  ground,  to  be  correct;  through  either  end  of  it,  say  at  its 
farther  end  2,  draw  a  short  meridian  line,  parallel  to  which  draw 
others  through  every  corner. 

Now,  having  the  bearing  of  side  2,  N  75°  32'  E,  and  requiring 
that  of  side  3,  it  is  plain  that  the  reverse  bearing  from  corner  2  is 
S  75°  32'  W,  and  that  therefore  the  angle  1  2  ra  is  75°  32', 
Therefore,  if  we  take  75°  32'  from  the  entire  corrected  angle  123, 
or  144°  57',  the  remainder  69°  25'  will  be  the  angle  m  2  3;  conse- 
quently the  bearing  of  side  3  must  be  S  69°  25'  E.  For  finding 
the  bearing  of  side  4,  we  now  have  the  angle  2  3  a  of  the  reverse 


PLANE   SURVEYING 


77 


bearing  of  side  3,  also  equal  to  69°  25',  and  if  we  add  this  to  the 
entire  corrected  angle  2  3  4,  or  to  69°  32'  we  have  the  angle  a  3  4 
=  69°  25'  +  69°  32'  =  138°  57',  which,  taken  from  180°,  leaves 
the  angle  I  3  4  =  41°  3' ;  consequently  the  bearing  of  side  4  must 
be  S  41°  3'  W. 

For  the  bearing  of  side  5,  we  now  have  the  angle  3  4  c  =  41° 
3',  which,  taken  from  the  corrected 
c  angle  3  4  5,  or  120°  43',  leaves  the 
angle  c  4  5  =  79°  40 ' ,  consequently 
the  bearing  of  side  5  must  be  N  79° 
40'  W.     At  corner  5,  for  the  bearing 
of  side  6,  we  have  the  angle  4  5  d  = 
D  79°  40',  which,  taken  from  133°  10', 
leaves  the  angle  d  5  6  —  53°  30'; 
consequently  the  bearing  of   side   6 
Fig.  61.  must  be  S  53°  30'  W:  and  so  with 

each  of  the  sides.  Nothing  but  careful  observation  is  necessary  to 
see  how  the  several  angles  are  to  be  employed  at  each  corner. 


FARM  SURVEYING. 

Method  of  Progression.  Farm  surveying  with  the  compass 
does  not  differ  in  any  essential  particular  from  the  methods 
outlined  for  surveying  a  series  of  lines.  If  the  boundary  lines  are 
irregular,  it  will  be  necessary  to  measure  offsets  at  proper  intervals, 
that  the  included  area  may  be  calculated.  The  method  above 
described  is  known  as  the  method  of  progression. 

Method  of  Radiation.  The  method  of  radiation  consists  in 
setting  up  the  instrument  at  some  point  inside  or  outside  the  field, 
from  which  all  the  corners  are  visible  and  accessible,  and  then 
measuring  the  bearing  and  lengths  of  the  lines  to  these  corners. 
Fig.  61  illustrates  the  method.  Set  up  the  compass  at  the  point  O, 
and  take  the  bearings  and  lengths  of  the  lines  O  A.  O  B,  O  C, 
O  D,  and  O  E. 

Method  of  Intersections.  Lay  off  a  base-line  of  convenient 
length  inside  or  outside  the  field,  from  which  all  the  corners  are 
visible.  Set  up  the  compass  at  one  end  of  the  base-line,  and  take 
the  bearings  from  it  to  each  corner  in  succession.  Remove  the 


85 


78  PLANE   SURVEYING 

compass  to  the  other  end  of  the  base-line,  and  take  the  bearings 
from  it  to  each  corner  in  succession.  Take  the  bearing  and  length 
of  the  base-line.  Now,  when  these  bearings  and  lengths  are 
plotted,  the  intersections  of  the  lines  will  define  the  corners. 

Proofs  of  Accuracy.  When  the  survey  of  a  field  is  plotted, 
if  the  end  of  the  last  course  meets  the  starting  point,  it  proves  the 
work,  and  the  survey  is  then  said  to  "close."  Errors  of  closure 
may  be  due  either  to  incorrect  lengths  of  lines  or  to  incorrect 
bearings,  or  to  both. 

Diagonal  lines  running  from  corner  to  corner  of  a  field  may 
be  measured  and  their  bearings  taken.  When  these  are  plotted, 
their  meeting  the  points  to  which  they  were  measured  proves  the 
accuracy  of  the  work. 

Finally,  the  accuracy  of  the  work  may  be  tested  by  calculating 
the  "latitudes  and  departures"  of  all  the  courses.  If  their  algebraic 
sum  is  equal  to  zero,  the  work  is  correct.  A  check  upon  the 
bearings  may  be  had  by  calculating  the  "deflection  angles"  between 
the  courses.  If  their  sum  is  equal  to  360  degrees,  the  bearings 
are  correct.  This,  however,  will  seldom  be  the  case.  A  certain 
amount  of  error  is  permissible,  depending  upon  the  nature  and 
importance  of  the  work. 

Field  Notes.  The  field  notes  may  be  recorded  in  various  ways, 
the  object  being  to  make  them  clear  and  full. 

1.  The  surveyor  may  make,  in  the  field  book,  a  rough  sketch 
of  the  survey  by  eye,  and  note  on  the  lines  their  bearings  and 
lengths.     If  a  protractor  and  scale  are  available,  the  actual  bearings 
and  lengths  of  the  lines  may  be  plotted  in  the  notebooks,  as  well 
as  offsets,  etc. 

2.  Draw  a  straight  line  up  the  page  of   the  notebook,  and 
record  on   it  the  bearings   and  lengths   of  the  lines.      Offsets, 
tie-lines,  etc.,  can  be  plotted  in  their  proper  positions. 

3.  Write  the    stations,    bearings,    and    distances   in  three 
columns.     This  method  has  the  advantage,  when  applied  to  farm 
surveying,  of  being  convenient  for  use  in  the  subsequent  calcula- 
tion of  contents,  but  does  not  give  facilities  for  noting  effects.     It 
is  illustrated  as  follows: 


PLANE  SURVEYING 


79 


STATIONS. 

BEARINGS. 

DISTANCES. 

0 

N.  32°      E. 

16.82 

1 

S.  36°      E. 

18.90 

3 
4 

S.  27^°  W. 
S.  16°      W. 

7.85 
15.30 

Notice  that  distances  are  given  in  Gunter's  chains,  and  in 
calculating  content  the  result  will  be  given  in  square  chains,  which 
can  be  reduced  to  acres  by  pointing  off  one  decimal  place. 

To  Change  Bearings.  In  certain  kinds  of  work  with  the 
compass,  it  is  convenient  to  assume  one  of  the  lines  as  a  meridian, 
and  it  then  becomes  necessary  to  change  the  bearings  of  all  of  the 
other  lines  to  conform  with  the  assumed  meridian.  This  case  is 
best  illustrated  by  an  example. 

The  bearings  of  the  sides  of  a  field  are  here  shown:  Suppose 
now  that  the  first  course  is  assumed  as  a  meridian,  that  is,  that  its 


STATIONS. 

BEARINGS. 

DISTANCES. 

1 

N.  35°      E. 

2.70 

2 

N.  83}£°  E. 

1.29 

3 

S.  57°      E. 

2.22 

4 

S.  34^°  W. 

3.55 

5 

N.  56^°  W. 

3.23 

bearing  is  due  north  and  south.     Required  the  bearings  of  the 
remaining  courses. 

Since  the  courses  are  changed  to  the  west  by  35°,  the  new 
bearing  of  course  2  will  be  N  48|°  E.  Of  course  3  it  will  be 
57°  +  35°  =  92°,  or  the  new  bearing  will  be  N  88°  E.  Of  course 
4  it  will  be  34^°  —  35°,  or  |°  in  the  next  quadrant,  or  the  bearing 
will  be  S  |°  E.  Of  course  5  it  will  be  56|°  +  35°  =  91£°,  or  the 
bearing  will  be  S  88|°  W. 

EXAMPLE    FOR   PRACTICE. 

The  bearings  of  a  series  of  courses  are  given  as  follows: 
The  bearing  of  the  first  course  is  changed  to  due  north  and  south. 


87 


80 


PLANE   SURVEYING 


It  is  required  to  determine  the  bearings  of  all  the  courses,  due 
to  this  change.     Find  bearings  and  plot  the  lines. 

Ans.     Course  2  =  N  62£°  E;  3  =  N  9°  W;  4  =  N  68°  W. 


STATIONS. 

BEARINGS. 

DISTANCES. 

1 

S.    21°    W. 

.  12.41 

2 
3 

N.  83*4°  E. 
N.  12°  E. 

5.86 
8.25 

4 

N.  47°  W. 

4.24 

Latitudes  and  Departures.  The  latitude  of  a  point  is  its 
distance  north  or  south  of  some  line  taken  as  a  parallel  of  latitude, 
or  line  running  east  and  west. 

The  longitude  of  a  point  is  its  distance  east  or  west  of  some 
line  taken  as  a  meridian,  or  line  running  north  and  south. 

The  distance  that  one  end  of  a  line  is  north  or  south  of  the 
other  end  is  the  "  Difference  of  Latitude"  of  the  two  ends  of  the 
line,  and  is  called  its  northing  or  southing,  or  its  latitude. 

The  distance  that  one  end  of  a  line  is  east  or  west  of  the 
other  end  is  the  "  Difference  of  Longitude  "  of  the  two  ends  of  the 
line,  and  is  called  its  easting  or  westing,  or  its  departure. 

The  terms  Latitude  Difference  and  Longitude  Difference  have 
of  late  come  into  quite  general  favor;  but  while  they  are  perhaps 
more  explicit,  they  are  certainly  cumbersome,  and  the  older  terms 
will  be  adhered  to  in  what  follows. 

In  Fig.  62,  N  S  represents  a  meridian, 
and  E  W  a  parallel  of  latitude.  If  we 
take  the  line  O  A,  its  bearing  as  given 
by  the  compass  is  the  angle  NOA.  The 
latitude  or  northing  of  the  point  A  is 
therefore  A  B  =  O  A  cos  NOA.  Its 
departure  or  easting  is  O  B  =  O  A  sin  NOA. 
To  find  the  latitude  of  a  course, 
multiply  the  length  of  the  course  by  the 
natural  cosine  of  the  bearing;  and  to  find 
the  departure  of  any  course,  multiply  the 
length  of  the  course  by  the  natural  sine  of  the  bearing. 


PLANE   SURVEYING 


81 


If  the  course  be  northerly,  the  latitude  will  be  north,  and  will  be 
designated  by  the  sign  +,  or  plus;  if  the  course  be  southerly,  the 
latitude  will  be  south,  and  will  be  designated  by  the  sign  — ,  or  minus. 

If  the  course  be  easterly,  the  departure  will  be  east,  and  will  be 
designated  by  the  sign  -+-,  or  plus;  if  the  course  be  westerly,  the 
departure  will  be  west,  and  will  be  designated  by  the  sign  — ,  minus. 

Thus  in  the  figure,  OA  is  of  plus  latitude  and  plus  departure; 
OP  is  of  plus  latitude  and  minus  departure;  OD  is  of  minus 
latitude  and  minus  departure;  and  OC  is  of  minus  latitude  and 
plus  departure. 

For  calculating  latitudes  and  departures,  a  set  of  traverse 
tables  may  be  procured;  but  a  table  of  natural  functions  will  be 
satisfactory',  though  possibly  less  convenient. 

Testing  a  Survey  by  Latitudes  and  Departures.  It  is  evident 
that  after  the  surveyor  has  gone  completely  round  a  field  or  farm, 
measuring  all  the  lengths  and  bearings,  returning  to  the  starting 
point,  he  has  gone  as  far  north  as  south,  and  as  far  east  as  west. 
In  other  words,  if  the  work  has  been  done  correctly,  the  algebraic 
sum  of  the  latitudes  must  equal  zero,  and  the  algebraic  sum  of  the 
departures  must  equal  zero.  This  condition,  however,  will  seldom 
be  attained,  and  it  becomes  necessary  to  decide  how  much  error 
may  be  permitted  without  necessitating  another  survey.  This  will 
depend  upon  the  nature  of  the  work  and  its  importance,  and  a 
surveyor  will  soon  determine  for  himself  his  factor  of  error, 
depending  partly  upon  his  instrument,  partly  upon  personal  skill, 
for  ordinary  cases.  If  it  is  necessary  to  depend  upon  a  "green"  hand 
to  carry  the  tape  or  chain,  this  may  prove  a  fruitful  source  of  error. 

We  shall  now  proceed  to  calculate  the  latitudes  and  departures 
of  the  survey  as  given  below.  Arrange  the  diagram  as  below  with 
seven  columns: 


STATIONS. 

BEARINGS. 

DISTANCES. 

LATITUDES. 

DEPARTURES. 

N. 

s. 

E. 

w. 

1 

S.  21°  W. 

12.41 

11.591 

4.443 

2 

3 

N.  83^°  E. 
N.  12°  E. 

5.86 
8.25 

0.691 
8.069 



5.819 
1.716 



4 

N.  47°  W. 

4.24 

•2.81)2 

3.104 

30.76 

11.652 

11.591 

7.535 

7.547 

89 


82  PLANE   SURVEYING 

The  cosine  of  the  bearing  of  course  1  is  0.934.12.41=11.591  —Latitude. 
The  sine  of  the  bearing  of  course  1  is  0.358.12.41=  4.443  —  Departure. 
The  cosine  of  the  bearing  of  course  2  is  0.118.5.86  =  0.691  +  Latitude. 
The  sine  of  the  bearing  of  course  2  is  0.993.5.86  =  5.819  +  Departure. 
The  cosine  of  the  bearing  of  course  3  is  0.978.8.25  =  8.069  +  Latitude. 
The  sine  of  the  bearing  of  course  3  is  0.208.8.25  =  1.716  +  Departure. 
The  cosine  of  the  bearing  of  course  4  is  0.682.4.24  =  2.892  +  Latitude. 
The  sine  of  the  bearing  of  course  4  is  0.732.4.24  =  3.104 —  Departure. 

The  latitudes  fail  to  balance  by  0.061  chains,  and  the  departures 
by  0.012  chains.  The  error  of  "closure"  of  the  survey  is  therefore 

I 2 — 

E  ==^.061  +  .012  =  0.062  +  chains,  or  approximately  4.09  feet. 

This  sum  may  be  divided  up  among  the  courses  in  proportion  to 
the  length,  or  the  bearings  may  be  corrected,  or  partly  one  and 
partly  the  other,  as  will  hereafter  be  explained. 

Balancing  the  Survey.  Before  proceeding  to  the  calculation 
of  the  content  of  a  field  or  farm,  the  survey  must  be  balanced; 
that  is,  the  latitudes  and  departures  must  be  corrected  so  that  their 
sums  shall  be  equal,  or  shall  balance.  As  to  whether  the  bearings 
or  lengths  shall  be  corrected,  will  depend  somewhat  upon  the 
conditions  tinder  which  the  survey  was  made.  If  the  surveyor  has 
reason  to  think  that  the  error  is  entirely  in  the  bearing  of  one  or 
more,  or  even  of  all  of  the  courses,  the  corrections  may  be  made 
accordingly.  If,  on  the  other  hand,  one  or  more  of  the  courses 
were  measured  over  difficult  ground,  it  may  be  presumed  that  the 
error  occurred  in  those  lines.  If,  however,  there  is  no  reason  to 
believe  that  one  course  is  in  error  more  than  another,  the 
differences  may  be  distributed  among  the  courses  in  proportion  to 
their  length,  according  to  the  following  proportions: 

As  the  length  of  any  course  is  to  the  sum  of  the  lengths  of  all 
the  courses,  so  is  the  correction  of  the  latitude  of  that  course  to 
the  total  error  in  latitude  of  all  the  courses. 

As  the  length  of  any  course  is  to  the  sum  of  the  lengths  of 
all  the  courses,  so  is  the  correction  of  the  departure  of  that  course 
to  the  total  error  in  departure  of  all  the  courses. 

The  practical  application  of  these  proportions  to  balancing  a 
survey  will  be  illustrated  from  the  preceding  problem: 

For  course  1 12.41  :  30.76  : ;  x  :  0.061 x  =  .0246,  correction  for  latitude. 

For  course  2. ...  5.86  :  30.76  ::  x  :  0.061. . .  .x  =  .0116,  correction  for  latitude. 


90 


PLANE  SURVEYING  83 

For  course  3. ...  8.25  :  30.76  : :  x  :  0.061. . .  .x  =  .0164,  correction  for  latitude. 
For  course  4. ...  4.24  :  30.76  ::  x  :  0.061. . .  .x  =  .0084,  correction  for  latitude. 

Since  the  sum  of  the  north  latitudes  is  the  greater,  the  corrections  will  be 
subracted  from  them  and  added  to  the  south  latitudes.  That  is  to  say,  the 
correction  for  course  1  will  be  added  to  11.591,  the  result  being  11.6156. 
The  correction  for  course  2  will  be  subtracted  from  0.691:  that  for  course  3 
will  be  subtracted  from  8.069;  and  so  on. 

For  course  1. .  .12.41  :  30.76  ::  x  :  0.012. .  .x  =  .0048,  correction  for  departure. 
For  course  2. . .  5.86  :  30.76  : :  x  :  0.012. .  .x  =  .0023,  correction  for  departure. 
For  course  3.. ..  8.25  :  30.76  ::  x  :  0.012..  ..x  =  .0032,  correction  for  departure. 
For  course  4.. ..  4.24  :  30.76  ::  x  :  0.012..  ..x  —  .0017,  correction  for  departure. 

The  corrections  are  to  be  subtracted  in  this  case  from  the  west  departure 
and  added  to  the  east  departure. 

In  this  example,  the  errors  are  small,  but  often  they  will  be  so 
large  as  to  raise  doubt  as  to  the  accuracy  of  the  survey.  In  such  a 
case,  go  carefully  over  all  the  computations,  and,  if  the  error  is  still 
too  large,  check  the  exterior  angles  of  the  figure  (their  sums 
should  equal  360°),  and  if  necessary  repeat  the  survey.  Having 
corrected  the  latitudes  and  departures,  the  corrected  bearings  of  the 
courses  may  be  deduced  from  the  trigonometric  ratio: 

corrected  departure 
Tan.  bearing  =  - 

corrected  latitude 

Calculating  the  Content.  After  a  field  has  been  surveyed,  its 
content  may  be  calculated  by  dividing  it  up  into  triangles,  trape- 
zoids,  etc.,  calculating  the  various  contents,  and  adding  them 
together.  This,  however,  is  at  best  a  cumbersome  method,  involv- 
ing much  work  of  calculation  and  great  chance  of  error.  The 
method  of  latitudes  and  departures  is  at  once  simple,  easily  applied, 
and  easily  checked. 

Before  proceeding  to  develop  a  formula  for  this  method,  it  will 
be  necessary  to  illustrate  and  define  certain  terms. 

Draw  a  line,  as  N  S  (Fig.  63),  through  the  extreme  east  or  west 
corner  of  the  field  for  a  meridian.  From  the  definitions  previously 
given,  the  difference  of  longitude  of  the  two  ends  of  a  line  is  the 
departure  of  the  line.  I  B  is  therefore  the  departure  of  the  line 
A  B.  The  departure  of  the  line  B  C  is  L  C;  that  of  E  F  is  S  F; 
and  that  of  A  F  is  O  Q. 

The  perpendicular  distance  of  each  station  from  the  given 
meridian  is  the  longitude  of  that  station,  plus  if  east,  minus  if 


fri 


84  PLANE  SURVEYING 


west.  Thus  the  longitude  of  A  is  zero;  that  of  B  is  I  B;  that  of  C  is 
I  B  +  L  C;  that  of  E  is  O  Q  +  F  S;  and  that  of  F  is  O  Q  = 
ZS  —  FS. 

The  difference  of  latitude  of  the  two  ends  of  a  line  is  called  the 

latitude  of  the  line.  Thus  the 
latitude  of  A  B  is  A  I;  that 
of  B  C  is  B  L;  that  of  E  F 
isES. 

The  distance  of  the  middle 
of  any  side  of  a  field  from  the 
meridian  is  called  the  longi- 
tude of  that  side.  Thus  the 
longitude  of  the  side  A  B  is 
GH;thatofBCisJX  =  GH 
+  K  M  +  MX;  and  that 
of  A  F  is  WV  =  OR  —  QR 

Fig.  63.  —  QP,  the  minus  signs  being 

used  in  this  instance  because  the  lines  E  F  and  A  F  bear  to 
the  west.  An  analysis  of  W  V  will  show  that  it  equals  O  R 
(longitude  of  preceding  course)  +  [ —  R  Q  (one-half  departure  of 
preceding  course)]  +  [ —  QP  (one-half  departure  of  the  course 
itself)]. 

To  avoid  fractional  quantities,  double  the  preceding  expres- 
sions and  then  deduce  a  general  rule  for  finding  double  longitudes. 
The  double  longitude  of  the  first  course  equals  the  departure 
of  that  course.  The  double  longitude  of  the  second  course  equals 
the  double  longitude  of  the  first  course,  plus  the  departure  of  the 
first  course,  plus  the  departure  of  the  second  course. 

The  double  longitude  of  any  course  equals  the  double  longi- 
tude of  the  preceding  course,  plus  the  departure  of  the  preceding 
course,  plus  the  departure  of  the  course  itself. 

We  shall  now  proceed  to  deduce  a  rule  for  determining  areas 
by  double  longitudes  and  departures,  and  shall  first  take  a  three- 
sided  field,  as  in  Fig.  64. 

Drawing  a  line  through  the  most  westerly  corner  A,  we  see 
that  the  area  of  the  field  will  be  the  difference  between  the  area  of 
the  trapezoid  D  B  C  M  and  the  combined  area  of  the  triangles 
D  B  A  and  A  C  M.  The  double  area  of  the  triangle  D  B  A  is  the 


PLAttE  SURVEYING 


85 


product  of  D  B  by  D  A,  or  the  double  longitude  of  A  B  by  the 
latitude  of  A  B.  The  resulting  product  will  be  north  or  plus. 
The  double  area  of  the  trapezoid  D  B  C  M  is  the  product  of  (D  B 
+  M  C)  =  2  G  H,  by  D  M,  that  is,  the  double  longitude  of  B  C  by 
its  latitude.  The  resulting  product  will  be  south  or  minus.  The 
double  area  of  the  triangle  ACM  will  be  the  product  of  M  C  by 


Fig.  64. 


Fig.  65. 


A  M,  or  the  double  longitude  of  the  course  A  C  by  its  latitude. 
The  resulting  product  will  be  north  or  plus.  Adding  together, 
then,  the  plus  products,  and  subtracting  from  the  minus  product, 
gives  as  the  result  the  double  area  of  the  field. 

We  shall  next  take  a  four-sided  field,  as  in  Fig.  65. 

It  is  evident  that  the  area  of  the  field  A  B  C  D  is  the  difference 
between  the  sum  of  the  areas  of  the  two  trapezoids  T  B  C  R 
and  RODE  and  the  sum  of  the  areas  of  the  triangles  A  B  T 
and  A  D  E. 

The  double  area  of  the  triangle  A  B  T  is  the  product  of  B  T 
by  A  T,  or  the  double  longitude  of  the  course  A  B  by  its  latitude. 
The  result  will  be  a  north  product  or  plus.  The  double  area  of  the 
trapezoid  T  B  C  R  will  be  the  product  of  (T  B  +  C  R)  =  2  L  P 
by  T  R — that  is,  the  double  longitude  of  the  course  B  C  by  its 
latitude.  The  result  will  be  a  south  product  or  minus.  The  double 
area  of  the  trapezoid  RODE  will  be  the  product  of  (R  C  +  D  E) 
=  2  F  K,  by  R  E,  or,  the  double  longitude  of  the  course  C  D 
by  its  latitude.  The  result  will  be  a  south  product  or  minus.  The 


PLANE  SURVEYING 


STA- 
TIONS 

BEARINGS 

DIS- 
TANCES 

LATITUDES 
N.          S. 

DEPARTURES 
E.          W. 

DOUBLE 
LONGI- 
TUDES 

DOUBLE  AREAS 

1 

S.  21°W. 

12.41 

11.616 

4.438 

—7.204 

83.682 

2 

5.86 

0.679 

5.821 

-5.821 

8.  062 

3 

X.  12°  E. 

8.25 

8.053 

1.719 

+  1.719 

13.843 

4 

X.  47°  W. 

4.24         2884 

3.102 

+0.336 

0.969 

98  494 
3.952 


47.271 
AREA=47.271  SQ.  Cns.=4  ACHES,  2  HOODS,  37  SQ.  PERCHES. 


double  area  of  the  triangle  A  D  E  will  be  the  product  of  E  D  by 
A  E,  or  the  double  longitude  of  the  course  A  D  by  its  latitude. 
The  result  will  be  a  north  product  or  plus.  Finally,  adding 
together  the  north  products,  adding  together  the  south  products, 
and  taking  the  difference  of  their  sums,  gives  as  the  result  the 
double  area  of  the  field  A  B  C  D. 

The  same  principle  will  apply  to  any  enclosed  area,  however 
great  the  number  of  the  sides.  The  area  will  always  be  one-half 
the  difference  of  the  sums  of  the  north  and  the  south  products 
arising  from  multiplying  the  double  longitude  of  each  course  Inj 
its  latitude. 

For  systematic  computation  arrange  the  work  as  follows: 

Arrange  the  columns  as  in  the  problem  on  page  83. 

Balance  the  latitudes  and  departures,  putting  the  corrected  quantities 
above  the  others  in  red  ink:  or  else  arrange  four  additional  columns,  and  enter 
them  in  their  proper  places. 

Compute  the  double  longitude  of  each  course  with  reference  to  a 
meridian  passing  through  the  extreme  east  or  west  station,  and  place  the 
results  in  another  column. 

Multiply  the  double  longitude  of  each  course  by  the  corrected  latitude 
of  that  course,  and  place  north  products  in  one  column  and  south  products  in 
another. 

Add  together  the  north  products  and  also  the  south  products,  ana  take 
the  difference  of  their  sums.  Divide  the  difference  by  two,  and  the  result 
will  be  the  area  desired. 

If  the  survey  has  been  made  with  a  Gunter's  chain,  the  result  will  be 
in  square  chains.  Divide  by  ten  to  reduce  to  acres. 


\\\ 


PLANE  SURVEYING  87 


To  test  the  correctness  of  the  calculation,  assume  the  meridian  as  passing 
through  the  extreme  station  upon  the  other  side  of  the  field,  and  carry  out 
the  work  in  detail  as  before. 

We  shall  now  proceed  to  calculate  the  content  of  the  field  given 
by  the  notes  on  page  81.  The  corrections  to  the  latitudes  will  be 
found  on  page  82,  and  the  corrected  departures  on  page  83. 

The  arrangement  of  the  columns  for  convenient  calculation  is 
as  described  on  page  86.  Upon  making  a  rough  sketch  of  the  courses, 
it  is  found  that  station  3  is  the  farthest  east;  and  therefore  the  double 
longitudes  will  be  calculated  beginning  with  course  3.  From  the 
definition  previously  given,  the  double  longitude  of  course  3  is  equal 
to  its  departure  =  +  1.719.  The  double  longitude  of  course  4 
equals  the  double  longitude  of  course  3,  plus  the  departure  of  course 
3,  plus  the  departure  of  course  4  -  1.719  +  1.719  +  (-  3.102)  = 
+  0.336.  The  double  longitude  of  course  1  equals  the  double  lon- 
gitude of  course  4,  plus  the  departure  of  course  4,  plus  the  departure 
of  course  1  =  0.336  +  (-  3.102)  +  (-  4.438)  =  -  7.204.  The 
double  longitude  of  course  2  equals  the  double  longitude  of  course  1, 
plus  the  departure  of  course  1,  plus  the  departure  of  course  2  =  + 
(-  7.204)  +  (-  4.438)  +  5.821  -  -  5.821.  Multiplying  these 
double  longitudes  by  their  respective  latitudes,  gives  the  quantities 
in  the  last  two  columns,  the  first,  third,  and  fourth  being  positive, 
and  the  second  negative.  Taking  the  difference  of  the  sums  of  the 
quantities  in  these  columns,  and  dividing  the  result  by  2,  gives  the 
content  of  the  field,  47.271  square  chains.  Dividing  by  10  gives 
4.7271  acres.  Reduce  to  roods  and  perches  by  multiplying  the 
decimal  part  by  4  and  40  successively. 

The  result  may  now  be  checked  by  beginning  with  the  most 
westerly  station,  and  it  will  be  necessary  to  recalculate  the  quantities 
in  the  last  three  columns. 

The  following  problems  are  taken  from  "Gillespie's  Surveying" 
(Staley): 

EXAMPLES    FOR    PRACTICE. 

Calculate  the  content  of  the  fields  from  the  data  tabulated 
below.  The  result,  where  found  in  square  metres,  should  be  reduced 
to  acres:  1  sq.  metre  =  .000247  acre. 


PLANE   SURVEYING 


(1) 


STATIONS. 

BEARINGS. 

DIST  \NCES. 

1 

N.34M"  E. 

2.73 

2 

N.  85°      E. 

1.28 

3 

S.  56%°  E. 

2.20 

4 

S.  34)4  °  W. 

3.53 

5                                     N.  56^°  W. 

3.20 

Ans.  1  acre,  0  roods,  14  perches. 


STATIONS.                                                 IJKARINOS. 

DISTANCES. 

1 

N.  35°     E. 

2.70 

2 

N.  83i£0  E. 

1  29 

3 

S.  57>      E. 

2.22 

1 

s.  3414°  w. 

3.55 

5 

N.  56^  "  W. 

3.23 

Ans.  1  acre,  0  roods,  15  perches. 

(3) 

STATIONS. 

BEARINGS. 

DISTANCES. 

1                                  S.    5   35'  W. 

2,388.88  meters 

2 

S.39°35'  W. 

1,060.27  meters 

3 

S.  50°  25'  E. 

3,078.31  meters 

4 

S.  79°    5'  E. 

325.00  meters 

5 

S.  53°  50'  E. 

275.00  meters 

6 

S.  48°  15'  W. 

200.00  meters 

7 

N.82°  45'  E. 

450.00  meters 

8 

S.  87°  40'  E. 

186.72  meters 

9 

N. 

8,768.12  meters 

10 

N.84°  25'  W. 

1,898.54  meters 

11 

S.    5°  35'  W. 

3,530.60  meters 

12 

N.  84  °  25  '  W.                        257  .  50  meters 

i     4,999      acres 

3      roods. 

39^  sq.  rods. 

Supplying  Omissions.     The  method  of  latitudes  and  depar- 
I'lres  may  be  applied  to  supplying  any  two  omissions  in  the  field 


PLANE  SURVEYING  89 

notes,  as  will  be  explained  in  connection  with  the  "Use  of  the 
Transit." 

Azimuth.  The  azimuth  of  a  line  is  the  horizontal  angle 
which  the  line  makes  with  some  other  line  taken  as  a  meridian. 
It  differs  from  bearing  in  that  it  is  measured  continuously  from  0° 
to  360°.  All  descriptions  of  property  must  be  given  in  terms  of 
bearings,  but  line  surveys  with  either  the  compass  or  the  transit 
had  better  be  given  in  terms  of  the  azimuth. 

In  astronomical  and  geodetic  work  it  is  customary  to  reckon 
azimuth  from  the  south  point  around  through  the  west,  through 


Fig.  66. 

360°.  For  the  ordinary  operations  of  surveying,  however,  it  is 
better  to  measure  the  azimuth  from  the  north  point  to  the  right 
through  360°. 

RESURVEYS. 

Where  the  boundary  lines  of  a  farm  or  town  have  been  obliter- 
ated and  the  corners  lost,  it  is  often  necessary  to  make 
resurveys  in  order  to  re-establish  them.  If  the  corners  can  be 
found  by  reliable  evidence,  they  must  be  accepted  as  corners  even 
though  the  second  bearings  and  lengths  of  the  lines  indicate 
different  points. 

It  sometimes  happens  that  some  corners  can  be  found  while 
others  cannot.  In  such  cases  a  series  of  random  lines  is  to  be  run 
with  the  old  bearings,  or  with  the  old  bearings  corrected  for  a 
change  in  declination  of  the  needle  between  the  two  dates. 

As  an  example,  let  the  records  in  an  old  deed  give  the  length 
and  bearings  of  three  lines  as  follows:  (See  Fig.  60.) 


90  PLANE  SURVEYING 

A6  N  60°  E  10  chains. 
be  N  45°  E  4  chains. 
cd  S  45°  E  8  chains. 

There  being  no  definite  data  at  baud  to  determine  the  change 
in  the  magnetic  declination  between  the  dates  of  the  two  surveys, 
the  lines  AB,  BC  and  CD  are  run  with  the  given  bearings  and 
distances  from  the  known  corner  A.  The  old  corners  I  and  c 
cannot  be  found;  but  on  arriving  at  D  the  old  comer  d  is  discov- 
ered at  a  point  20.4  links  S  and  12°  "W  from  D  It  is  required  to 
locate  the  old  corners  J  and  c. 

By  the  method  of  latitudes  and  departures  explained  before, 
the  lengths  of  the  lines  DA  and  (IA.  may  be  computed.  They  are: 
for  DA,  south  82°  47',  west  17.29  chains;  for  dA,  south  83°  26', 
west  17.22  chains. 

Now  the  error  Dd  between  the  two  corners  is  due  to  two 
causes:  (1)  the  continued  variation  in  the  magnetic  bearings  of  the 
old  surveys,  (2)  the  difference  in  the  length  of  the  chains  used. 
The  first  cause  alters  the?  polygon  AbcdA.  around  the  point  A  by  a 
small  angle.  The  second  cause  alters  the  length  of  the  sides  in  a 
constant  ratio.  The  difference  between  the  bearings  DA  and  t?A 
is  the  constant  angle,  while  the  ratio  of  the  length  of  the  old  lines 
is  the  constant  ratio.  To  find  the  bearings  of  the  old  line,  there- 
fore, each  of  the  given  bearings  is  to  be  corrected  by  the  amount 
83°  26'  minus  82°  47'  =  0°  39'.  To  find  the  length  of  the  old 

17  22 
line  each  of  the  given  lengths  is  to  be  multiplied  by  -     '"  =  0.99(5. 

Suppose  now  that  the  work  of  computation  has  been  done 
with  such  precision  that  the  error  in  chaining  must  be  regarded  as 
lying  in  the  old  survey.  Applying  these  results,  we  find  the 
adjusted  bearings  and  lengths  of  the  old  line  to  be, 

Ah  =  N  60°  39'  E  9.%  chains. 

he   =  N  45°  39'  E  3.99  chains. 

cd  =  S  44°  21'  E  7.97  chains. 

With  the  new  data  the  line  may  be  rerun  and  the  corners  I  and  c 
located,  a  check  on  the  field  work  being  that  the  lost  line4  should 
end  exactly  at  (L 

It  is,  however,  not  difficult  to  compute  the  length  and  bearings 
of  B  h  and  C  r,  so  that  I  and  c  may  be  located  from  the  points 
B  and  C. 


PLANE  SURVEYING  91 

Since  the  angle  B  A  b  is  small,  the  triangle  B  A  b  may  be  con- 
sidered similar  to  the  triangle  D  A  d.  We  will  then  have  the  pro- 
portion : 

Bb:Dd::AB:AD. 


A  similar  proportion  may   be   written   for   the   side   B  c,   and    the 
result  added  to  the  value  of  B  6. 

The  same  principle  .may  be  used  to  determine  the  bearings  of 
B  b  and  C  c,  so  that,  with  the  lengths  and  bearings  of  these  lines 
determined,  the  most  probable  location  of  the  old  corners  b  and  c 
can  be  fixed. 

EXAMPLE    FOR    PRACTICE. 

The  records  of  an  old  survey  read  as  follows: 

"Commencing  at  a  point  marked  No.  5  and  running  X  62°  E 
14  chains  to  a  stake  marked  A,  thence  running  N  432°  E  8.00  chains 
to  a  stake  marked  B,  thence  N.  5°  W.  12.00  chains  to  a  stake  C, 
thence  N  72£°  E.  10.25  chains  to  a  stake  D,  thence  S  12°  W  6.43 
chains  to  a  stake  marked  No.  3.  On  running  the  lines,  the  end 
of  the  last  one,  instead  of  being  at  a  stone  marked  No.  3,  was  0.62 
chain  due  E  from  it."  Find  the  adjusted  bearings  and  lengths 
of  the  old  lines;  also  find  the  distance  and  direction  from  each  station 
of  the  new  survey  to  the  corresponding  corner  of  the  old. 

DIVISION    OF    LAND. 

The  method  of  latitudes  and  departures  is  especially  useful  in 
the  division  of  land.  The  problem  is  usually  as  follows:  Given  the 
lengths  and  bearings  of  the  sides  of  a  field  containing  a  certain  area; 
it  is  required  to  divide  the  area  into  certain  parts  by  a  line  running  in 
a  certain  direction,  in  which  case  it  is  necessary  to  determine  the 
starting  point  of  the  dividing  line.  Or  it  is  required  that  the  line 
shall  begin  at  a  certain  point,  in  which  case  it  is  necessary  to  deter- 
mine the  direction  of  the  line. 

A  certain  field  is  described  as  follows: 

1  N.  63°  51'  W.          6.91  Chs. 

2  N.  63°  44'  W.          7.26     " 

3  N.  69°  35'  W..         3.34     " 


92 


PLANE  SURVEYING 


4 

N.  77°  50'  W. 

6.54  Chs. 

5 

N.  31°  24'  E. 

14.38     " 

6 

N.  31°  18'  E. 

16.81     " 

7 

S.  68°  55'  E. 

13.64     " 

8 

S.  68°  42'  E. 

11.54     " 

9 

S.  33°  45'  W. 

31.55     " 

Beginning  at  a  point  upon  the  side  91,  it  is  required  to  divide 
the  area  into  two  equal  parts  of  37  acres  by  a  line  having  a  bearing  of 
N.  68°  46'  W.  Some  preliminary  calculations  will  be  necessary  to 

determine  the  start- 
ing point  of  the 
dividing  line  such 
that  the  resulting 
area  will  be  nearly 
that  desired,  and 
this  dividing  line 
will  be  practically 
parallel  to  a  line 
connecting  stations 
7  and  9.  Call  the 
dividing  line  A  B 
(see  Fig.  67),  and 
if  this  line  begins 
at  a  point  15  chains 
from  station  9,  the 

67'  length  of  A  B  will 

be    24.528    chains; 

that  of  B  7  will  be  14.90  chains.  We  shall  now  proceed  to  calculate 
the  area  by  latitudes  and  departures,  beginning  with  station  9. 
The  calculations  are  shown  in  the  following  table,  and  the  result, 
36.5186  acres,  is  short  of  the  required  area  by  0.4814  acre  or  4.814 
square  chains. 


STATION 

BEARINGS 

DlS-               IvATIl 

TANCE8            N. 

PUDE8 

S. 

JJEPAI 

E. 

<™RES    LoNQI. 

TUPKS 

DOUBLE  AREAS 

9 
A 
B 

7 
8 

S.  33°  45'  W. 
N.  68°  46'  W. 
N.  31°   18'  E. 
S.  69°    55'  E. 
S.  68°    42'  E. 

15.00 

24  528 
14  90 
13.64 
11.54 

8  883 
12.731 

12.472 

4.907 
4  192 

7  741 
12.727 
1(1  7.VJ 

8.334 
22  863 

—  8  334 
-39531 
—54  653 
—34  185 
-10  706 

103.942 

167.746 
44.880 

351.154 
695.787 

AREA  =  36.5186  ACRES. 


316.568  1046.941 

316.568 

2  |  730.373 

865.186 


PLANE  SURVEYING 


Our  line  A  B  must  therefore  be  moved  farther  south  such  a 
distance  that  the  area  enclosed  between  the  first  and  second  posi- 
tions shall  equal  4.814  square  chains.  Considering  this  area  as  a 
rectangle  (which  it  will  be  nearly),  the  line  A  B  will  be  moved  south 

'  0     =  0.1961  chain,  or,  say,    0.2    chain.     The  line  9/1    will 
^4.o^o 

be  therefore  15.2  chains  in  length.  The  length  of  A  B  will  not  be 
materially  changed.  B  7  will  now  be  15.1  chains  in  length,  while 
78  and  89  will  be  the  same  as  before.  The  latitudes  and  depar- 
tures of  9  A,  A  B,  and  B  7  must  be  recalculated,  as  well  as  the 
double  longitudes  of  all  of  the  courses.  The  calculations  are  found 
in  the  following  table,  and  there  results  an  area  equal  to  37.015 
acres. 


STATION 

i 
BEARINGS  |        |s~ 

NC 

LATITUDES 

N.              S. 

DEPARTURES 

E.             W. 

DOUBLE 
LONGI- 
TUDES 

DOUBLE  AREAS 

9 

S.  33°45'W. 

15.20 

12.638 

8.445     —  8.445       106.728 

A 

N.  68°  46'  W. 

24.528 

8.883 

22.863     —39.753 

353.126 

B 

N.31°18'E. 

15.10 

12.902 

7.845 

-  54.771 

706.655 

7 

S.68°55'E. 

13.64 

4.907 

12.727                    -34.199 

167.814 

8 

S.  68°  42'  E. 

11.54 

4.192 

10.752 

-10.720 

44.938 

319.480       1059.781 

319.480 

2  774U.30I 

370.150 


AREA  =  37.015  ACRES. 


Referring  to  Fig.  67,  starting  at  a  point  A  on  91,  12.25  chains 
from  station  9,  it  is  required  to  find  the  length  and  bearing  of  A  B 
such  that  the  area  9  A  B  789  shall  equal  32  acres. 

First  draw  a  line  from  A  to  station  7,  and  by  latitudes  and 
departures  calculate  the  area  A  7  89  A,  and  determine  the  length  and 
bearing  of  A  7.  Call  this  latter  area  H .  Then  the  area  A  El  A 
must  equal  32-H.  From  the  point  B,  erect  a  perpendicular  B  C 


to  the  line  A  7.  The  urea  of 


B  7  A  will  equal '     '  X  B  C 


(32- 


PLANE  SURVEYING 


H);  .'.  B  C  = 


2  (32-g) 

A7      ' 


B  C 

Therefore  B  7  =  - — n-;r>v  In  the  tri- 
sin  B  <  C 


angle  A  B  7  A  we  now  have  two  sides  and  the  included  angle  from 
which  to  calculate  the  length  and  bearing  of  A  B. 

THE  TRUE  MERIDIAN. 

In  order  to  ascertain  the  true  meridian  of  a  given  place, 
several  methods  may  be  pursued.  The  general  practice  is  to  use 
the  star  Polaris  at  culmination  or  elongation.  This  star  is  on  the 
meridian,  nearly,  when  a  plumb  line  covers  it  and  the  star  Zeta, 


^      DELTAV 

'--    I  ' 

^CASSIOPEIA   / 


Fig.  68. 

the  next  to  the  end  of  the  handle  of  ''The  Dipper."     See  Fig.  68. 

When   Polaris   is    on    the    meridian,   as   illustrated    in   this 

instance,  it  is  said  to  be  at  "culmination."     This  star  is  often 


PLANE  SURVEYING  95 

referred  to  as  the  north  or  pole  star.  It  is  about  1^°  from  the 
pole,  and  revolves  around  the  pole  once  every  23  hours  and  56 
minutes.  Thus  it  is  apparent  that  it  comes  on  the  true  meridian 
twice  each  day.  The  arrows  in  the  figure  indicate  the  direction  of 
the  rotation. 

To  Determine  the  True  Meridian  by  the  Compass.  With 
Polaris  at  Eastern  or  Western  Elongation.  To  determine  the 
true  meridian  by  means  of  the  compass,  take  a  plumb-line,  and 
attach  one  end  of  the  line  to  any  suitable  support  situated  as  far 
above  the  ground  as  practicable,  so  as  to  have  a  clear  field  of  view 
about  20  feet  away.  A  board  nailed  on  a  telegraph  pole,  tree,  or 
post  at  right  angles,  will  suffice  for  this  purpose.  The  plumb-bob 
maybe  of  any  suitable  material,  of  about  5  Ibs.  in  weight,  as  a  brick, 
stone,  iron  ring,  or  coupling.  It  will  serve  the  same  purpose,  with 
as  accurate  results,  as  the  most  highly  polished  or  carefully  manu- 
factured plumb-bob.  The  plumb-line  should  be  about  25  feet  in 
length,  depending  upon  the  latitude  of  the  place,  since  the  altitude 
of  the  pole  above  the  horizon  at  any  place  is  equal  to  the  latitude 
of  that  place. 

Illuminate  the  plumb-line  just  below  its  support  by  means  of 
a  bull's-eye  lantern,  lamp,  or  candle,  care  being  taken  not  to 
obliterate  the  line  from  the  view  of  the  observer.  The  best  way  is 
to  screen  the  light,  and  throw  the  light  on  the  plumb-line  by 
means  of  a  reflector. 

Next  unfasten  one  of  the  uprights  of  the  compass,  and  place  it 
on  a  horizontal  rest  at  some  convenient  point  south  of  the  plumb- 
line,  say  30  feet  in  an  east  or  west  direction,  and  in  such  a 
position  that  when  viewed  through  the  peep-sight,  Polaris  will 
appear  about  two  feet  below  the  support  of  the  plumb-line.  It  is 
customary  to  determine  this  position  by  trial  the  night  before 
the  observation. 

About  25  minutes  before  the  time  of  elongation,  as  per  table 
on  page  130,  bring  the  peep-sight  into  the  same  line  of  sight  with 
the  plumb-line  and  the  star  Polaris.  Before  reaching  elongation, 
the  star  will  move  away  from  the  plumb-line,  to  the  east  for  eastern 
elongation,  and  to  the  west  for  western  elongation.  Hence,  by 
moving  the  peep-sight  in  the  proper  direction— that  is,  east  or 
west — the  star  can  be  kept  on  the  plumb-line  until  it  appears  to 


PLAtfE  SURVEYING 


remain  stationary,  thus  indicating  that  it  has  reached  its  point  of 
elongation.  The  peep-sight  will  now  be  secured  in  place  by  a 
clamp  or  weight,  with  its  exact  position  marked  on  the  rest.  Now 
defer  all  further  operations  until  the  next  day. 

The  next  morning  place  a  slender  flag  or  ranging  pole  at  a 
distance  of  200  or  BOO  feet  from  the  peep-sight,  and  exactly  in  line 
with  the  plumb-line.  Next  carefully  measure  this  distance,  and 
take  from  the  table  (page  130)  the  azimuth  of  Polaris,  corresponding 
to  the  latitude  of  the  station  of  observation;  find  the  natural 
tangent  of  this  azimuth,  and  multiply  it  by  the  measured  distance 
from  the  peep-sight  to  the  rod.  The  product  will  express  the 
distance  to  be  laid  off  from  the  rod,  exactly  at  right  angles  to  the 
direction  already  determined  —  that  is,  to  the  west  for  eastern 
elongation,  and  to  the  east  for  western  elongation;  and  this  point 
with  the  peep-sight,  will  define  the  direction  of  the  meridian  with 
sufficient  accuracy  for  the  needs  of  local  surveyors. 

The  position  of  the  pole  star  may  be  found  by  means  of  the 
two  stars  /3  and  a  in  the  bowl  of  the  "The  Dipper'1  (Fig.  68),  which 
are  called  the  "pointers"  because  of  their  pointing  approximately 
to  the  pole  star. 

THE   TRANSIT. 

Construction.  The  transit  is  used  for  measuring  horizontal 
and  vertical  angles  directly,  and  for  measuring  bearings  indirectly. 
It  consists  of  a  telescope  mounted  in  standards  attached  to  a  divided 
horizontal  plate,  the  telescope  serving  to  define  accurately  the  line 
of  sight;  while  the  horizontal  plate,  divided  into  degrees,  minutes, 
and  twenty  or  thirty  seconds  of  arc,  makes  it  possible  to  measure 
small  horizontal  angles.  The  instniment  is  provided  with  a  three- 
or  four-screw  leveling  base,  by  means  of  which  it  is  attached  to 
the  tripod. 

The  telescope  is  similar  in  construction  to  that  of  the  Wye 
level,  but  is  shorter  and  of  less  magnifying  power,  a  power  of  from 
24  to  26  diameters  being  about  the  average  for  the  ordinary  transit. 
The  eye-piece  may  be  either  inverting  or  erecting,  but  the  former 
is  to  be  preferred. 

Since  the  principal  function  of  the  transit  is  to  secure  align- 
ment, the  telescope  must  be  capable  of  movement  in  a  vertical 


104 


PLANE   SURVEYING  97 

plane,  and  to  that  end  is  supported  in  the  standards  by  a  transverse 
axis,  permitting  the  telescope  to  be  "transited,"  that  is.  turned 
through  a  complete  vertical  circle. 

For  measuring  horizontal  angles  the  instrument  is  arranged 
with  an  upper  and  a  lower  motion,  sometimes  called  the  upper  and 
the  lower  "limb."  The  lower  limb  is  supported  by  the  leveling 
base  by  means  of  a  hollow  conical  axis;  and  into  it  is  fitted,  in  turn, 
the  conical  axis  of  the  upper  limb.  Each  limb  may  be  turned 
independently  of  the  other,  or  they  may  be  clamped  together  and 
to  the  leveling  base.  The  lower  limb  carries  the  divided  circle 
and  the  upper  limb  the  vernier.  For  ordinary  purposes  the 
circle  is  divided  to  one-half  degrees,  and  reads  to  single  minutes 
by  means  of  the  vernier.  It  may  also  be  divided  so  as  to  read 
to  20  or  30  seconds,  and  occasionally  to  10  seconds.  The  divisions 
of  the  circle,  however,  should  not  be  so  crowded  as  to  render  the 
reading  difficult,  and  the  graduations  should  be  properly  adjusted 
to  the  magnifying  power  of  the  telescope. 

The  verniers  may  be  set  at  right  angles  to,  or  parallel  with  the 
line  of  sight,  or  at  30°  thereto.  With  the  verniers  parallel  with 
the  line  of  sight — that  is  to  say,  directly  under  the  telescope — or 
making  an  angle  of  30°  with  the  line  of  sight,  the  observer 
can  read  the  angles  without  moving  from  his  position,  thereby 
avoiding  the  risk  of  disturbing  the  instrument  by  walking  around 
it.  See  Fig.  69. 

For  leveling  the  instrument,  there  are  provided  two  level 
tubes  set  at  right  angles  to  each  other.  These  are  shown  in  the 
figure.  One  of  them  is  attached  to  the  upper  plate,  while  the  other 
may  be  attached  either  to  the  upper  plate  or  to  one  of  the  standards. 
On  account  of  lack  of  space  these  level  tubes  are  quite  short. 

The  four-screw  leveling  base  may  consist  of  two  parallel  plates 
connected  to  each  other  by  a  one-half  ball  and  socket  joint, 
or  the  upper  plate  may  be  replaced  by  a  ribbed  casting.  The 
four  leveling  screws  rest  in  cups  upon  the  lower  plate  and 
extend  through  the  upper  plate  or  casting.  The  leveling  base 
is  attached  to  the  instrument  proper,  and  the  whole  is  attached 
to  the  tripod  by  screwing  to  a  casting  firmly  attached  to  the 
legs.  The  vertical  axis  is  furnished  with  a  hook,  to  which  may 
be  attached  a  plumb-line  for  the  accurate  centering  of  the  instru- 


9S 


PLANE   SURVEYING 


ment.  A  shifting  center  is  also  provided,  by  means  of  which,  after 
the  instrument  has  been  approximately  centered  over  a  stake,  it 
may  be  accurately  adjusted  by  loosening  the  leveling  screws  and 
shifting  the  instrument  upon  the  lower  leveling  base.  See  Fig.  69. 


Fig.  69. 

The  three-screw  leveling  base  is  necessarily  larger  and  differs 
in  detail  from  the  four-screw.  The  upper  plate  carrying  the  screws 
is  permanently  attached  to  the  instrument;  and  the  lower  ends  of 


106 


U.  S.  GEOLOGICAL  SURVEY  CAMP  ON  CHOPAKA  MOUNTAIN.  WASHINGTON 


PLANE   SURVEYING 


99 


the  screws  rest  upon  the  tripod  casting,  to  which  it  is  attached  by 
a  single  center  screw  fitted  with  a  strong  spiral  spring  that  engages 
upon  a  thread  cut  upon  the  vertical  axis  of  the  instrument.  See 
Fig.  72. 


Fig.  70. 

The  four-screw  base  commends  itself  from  the  fact  that  it  can 
quickly  be  leveled  approximately;  and,  no  matter  how  much  the 
threads  are  worn,  the  instrument  can  be  brought  to  a -sol  id  bearing. 


107 


100 


PLANE  SURVEYING 


The  three-screw  base,  however,  is  more  easily  manipulated,  and  all 
danger  of  binding  the  screws  and  springing  the  plates  is  obviated. 
Whichever  type  of  instrument  is  preferred,  the  screws  should  work 


Pig.  71. 

smoothly  and  evenly,  and  the  pitch   should  be   adjusted  to  the 
sensitiveness  of  the  bubbles. 

Most  transits  are  fitted  with  a  compass  set  in  the  upper  plate 


108 


PLANE   SURVEYING 


101 


Fig.  72. 

between  the  standards;  but  for  city  work,  triangiilation,  etc.,  the 
compass  is  dispensed  with. 


109 


102 


PLANE   SURVEYING 


For  measuring  vertical  angles,  the   transit  is  fitted  with   a 
vertical  arc  or  circle  divided  usually   to  one-half  degrees,  and 


Pig.  73. 

reading  to  single  minutes  by  the  vernier.    A  level  tube  may  also  be 
attached   to  the   under   side  of   the   telescope;  and  when  this  is 


110 


PLANE   SURVEYING  103 

provided  the  transit  may  be  used  as  a  leveling  instrument. 
A  striding  level  resting  upon  the  standards  may  also  be  provided, 
by  means  of  which  the  instrument  can  be  more  accurately  leveled 
than  by  the  short  levels  upon  the  upper  limb.  See  Fig.  70. 

The  telescope  should  always  be  provided  with  stadia  wires, 
either  fixed  or  adjustable,  though  the  former  are  preferable.  See 
article  on  "Stadia." 

The  gradienter  screw  is  a  device  attached  to  the  clamp  of  the 
telescope,  by  means  of  which  grades  can  be  established,  and 
horizontal  distances,  vertical  angles,  and  differences  of  level  can  be 
measured  with  great  rapidity.  See  article  on  "  Gradienter"  in 
Part  III. 

Surveyor's  Transit.  This  instrument  is  the  plain  transit, 
capable  usually  of  measuring  horizontal  angles  only,  but  occasion- 
ally fitted  with  a  vertical  circle  or  arc  for  measuring  vertical  angles. 
See  Fig.  69. 

Engineer's  Transit.  When  the  instrument  is  provided  with  a 
vertical  circle  or  arc,  a  level  underneath  the  telescope,  with  or 
without  gradienter  screw,  it  is  called  the  engineer's  transit.  See 
Fig.  70. 

Tachymeter.  This  term,  meaning  rapid  measurer,  has  of 
recent  years  been  applied  to  an  instrument  having  a  level  attached 
to  the  telescope,  a  vertical  arc  or  circle,  and  stadia  wires.  Such 
an  instrument  is  adapted  to  the  rapid  location  of  points  in  a 
survey,  since  it  is  capable  of  measuring  the  three  cp-ordinates  of  a 
point  in  space,  i.  e.,  the  angular  co-ordinates  of  altitude  and 
azimuth,  and  the  radius-vector  or  distance.  The  compass  and 
gradienter  are  auxiliaries  in  the  measurement  of  angles;  arid  an 
instrument  having  them  in  addition  to  the  essential  features 
mentioned  above,  is  more  perfectly  adapted  for  tachymetric  work. 
See  Fig.  71. 

Theodolite.  This  term  is  applied  to  an  instrument  so  con- 
structed that  the  telescope  will  not  transit,  but,  in  order  to  take 
backward  sights,  the  telescope  must  be  lifted  out  of  its  supports 
and  turned  end  for  end.  See  Fig.  73. 

Transit-Theodolite.  This  name  is  applied  to  an  instrument 
in  which  the  telescope  not  only  can  be  transited,  but  also  lifted 
out  of  its  supports  and  turned  end  for  end.  See  Fig.  72. 


Ill 


104  PLANE  SURVEYING 

Adjustment.  When  used  merely  as  an  angle-measurer,  the 
following  adjustments  should  be  tested  and,  if  necessary,  corrected: 

1st.  To  ascertain  if  the  bubble  tubes  are  perpendicular  to  the 
vertical  axis  of  the  instrument. 

To  test,  attach  the  instrument  to  the  tripod  and  "set  up"  firmly 
on  solid  ground,  preferably  shaded  from  sun  and  wind.  Revolve 
the  transit  upon  its  vertical  axis  so  as  to  bring  the  bubble  tubes 
parallel  to  a  pair  of  diagonally  opposite  leveling  screws.  Bring  the 
bubble  of  one  of  the  tubes  to  the  center  by  means  of  these  screws. 
Do  the  same  with  the  second  bubble  tube.  Adjusting  the  second 
tube  will  throw  the  first  one  out,  but  repeat  the  alternate  operations 
until  each  bubble  stands  in  the  center  of  its  tube.  Now  revolve 
the  instrument  upon  its  vertical  axis  through  180°,  and  note  if  the 
bubble  of  each  tube  still  stands  in  the  center.  If  so,  the  tubes  are 
in  adjustment. 

If  the  bubble  of  either  tube  runs  to  one  end,  bring  it  half-way 
back  to  the  center  by  raising  the  opposite  end  of  the  tube  by  means 
of  the  capstan-headed  screw.  Relevel  the  instrument  by  the 
leveling  screws,  and  again  test  the  tubes.  Repeat  the  operation, 
until  the  bubbles  stand  in  the  centers  of  the  tubes  in  all  positions  of 
the  instrument.  It  is  advisable  to  carry  out  this  adjustment  as 
precisely  as  possible,  as  it  will  facilitate  the  remaining  adjustments. 
If  after  several  trials,  it  is  found  impossible  to  adjust  the  bubbles 
to  the  centers  of  the  tubes,  either  the  vertical  axis  is  bent  or  the  plates 
are  sprung,  and  the  instrument  should  be  sent  to  the  maker  for 
correction.  If  one  tube  adjusts  and  the  other  does  not.  the  fault  is 
in  the  tube,  and  a  new  one  should  be  ordered. 

2d.  To  make  the  line  of  collimation  revolve  in  a  plane,  or, 
in  other  words,  to  make  the  line  of  collimation  perpendicular  to 
the  horizontal  axis  of  the  telescope* 

To  test,  having  made  the  first  adjustment,  level  the  instrument 
carefully  and  clamp  the  upper  limb.  Drive  a  stake  into  the  ground 
about  300  feet  ahead  of  the  instrument,  and  drive  a  tack  in  the 
head  of  the  stake.  By  means  of  the  lower  motion  revolve  the 
instrument  on  its  vertical  axis  until  the  intersection  of  the  cross- 
hairs approximately  covers  the  tack.  Now  clamp  the  lower  motion, 
and  carefully  adjust  the  line  of  sight  upon  the  tack  by  means  of 
the  lower  tangent  screw.  Without  disturbing  either  the  upper 


112 


PLANE  SURVEYING  105 

or  the  lower  limb,  transit  the  telescope,  that  is,  revolve  it  vertically, 
and  sight  to  a  tack  in  the  head  of  a  stake  driven  into  the  ground 
about  300  feet  behind  the  instrument.  Carefully  adjust  the  tack 
to  the  intersection  of  the  cross-hairs.  Now  unclamp  the  lower 
motion,  and  revolve  the  instrument  upon  its  vertical  axis  until  the 
intersection  of  the  cross-hairs  again  covers  the  tack  in  the  first 
stake.  Clamp  the  lower  motion,  adjust  the  line  of  sight  carefully 
by  means  of  the  tangent  screw,  again  transit  the  telescope,  and 
sight  in  the  direction  of  the  second  stake.  If  the  intersection 
of  the  cross-hairs  falls  upon  the  tack  in  the  second  stake,  the  line 
of  collimation  is  in  adjustment.  If  it  does  not,  it  will  have  to 
be  adjusted. 

In  Fig.  74,  A  is  the  position  of  the  instrument,  and  B  is  the 
forward  stake.  If  the  instrument  is  in  adjustment,  the  line  of 
sight  after  transiting  the  telescope  and  revolving  upon  the  vertical 
axis  should  strike  the  point  B'.  If  the  instrument  is  not  in 


Fig.  74. 


adjustment,  the  line  of  sight  after  transiting  the  telescope  will 
in  the  first  instance  strike  some  point  as  C.  Drive  a  stake  at  this 
point  and  carefully  center  a  tack.  After  revolving  the  instrument 
upon  its  vertical  axis  and  again  transiting  the  telescope,  the  line 
of  sight  will  fall  at  a  point  C'  as  far  on  one  side  of  B'  as  C  was 
on  the  other.  Drive  a  stake  at  C '  and  carefully  center  it.  Carefully 
measure  the  distance  C  C ' ;  and  center  a  stake  at  B ' ,  half-way 
between  the  two  points.  Now,  by  means  of  the  capstan-headed 
screws  attached  to  the  diaphragm  carrying  the  cross-hairs,  move  the 
cross-hairs  until  their  intersection  covers  the  point  B",  midway 
between  B'  and  C'.  Now  repeat  the  operation  of  testing  the 
adjustment  and  correcting  the  position  of  the  line  of  collimation, 
until  the  points  B  and  B '  are  in  the  same  straight  line. 

It  is   necessary   only   that   the   line   of  collimation  shall  be 
accurately  in  adjustment;  but  for  convenience  in  using  the  transit 


113 


106  PLANE   SURVEYING 

as  an  angle-measurer  it  is  desirable  that  the  vertical  cross-hair  be 
at  right  angles  to  the  horizontal  axis  of  the  telescope  when  the 
instrument  is  level. 

To  test  this,  set  up  the  instrument  at  some  convenient  point, 
200  or  300  feet  from  a  wall,  tree,  or  other  convenient  object,  upon 
which  a  point  is  clearly  denned  by  a  tack  or  otherwise.  Carefully 
level  the  instrument,  and  cover  this  point  accurately  with  the  lower 
extremity  of  the  vertical  hair.  Clamp  the  horizontal  axis  of  the 
telescope;  and  by  means  of  the  tangent-screw  slowly  move  the 
telescope  in  a  vertical  plane,  and  note  if  the  hair  continues  to  cover 
the  point  from  one  extremity  to  the  other.  If  it  does,  the  hair  is  in 
its  proper  position.  If  riot,  loosen  the  diaphragm  screws  and  turn  the 
diaphragm  vertically  until  the  hair  covers  the  point  from  end  to 
end.  This  adjustment  will  disturb  the  last  one  and  the  two  must 
be  tested  and  corrected  alternately  until  in  perfect  adjustment. 

If  the  transit  is  to  be  used  for  leveling,  it  is  necessary  that  the 
horizontal  cross-hair  be  in  the  optical  center  of  the  object  glass. 

To  test,  set  the  instrument  up  firmly  200  or  300  feet  from  a 
wall,  tree,  or  other  convenient  object,  and,  after  leveling,  carefully 
center  the  intersection  of  the  cross-hairs  upon  a  well-defined  point. 
Clamp  the  axis  of  the  telescope,  turn  the  instrument  upon  its 
vertical  axis  through  180°,  and  carefully  center  a  point  upon 
the  intersection  of  the  cross-hairs  in  this  new  position.  Clamp  the 
vertical  axis,  unclamp  the  telescope  axis,  transit  the  telescope,  and 
carefully  center  the  intersection  of  the  cross-hairs  upon  the  first 
point.  Now  clamp  the  telescope,  loosen  the  vertical  axis,  and  again 
revolve  the  instrument  through  180°.  If  the  line  of  collimation 
again  strikes  the  second  point,  the  horizontal  cross-hair  is  in 
adjustment.  If  not,  carefully  center  this  third  point,  bisect  the 
distance  between  the  second  and  third  points,  and  move  the  cross- 
hair diaphragm  until  the  intersection  of  the  cross-hairs  covers 
this  fourth  point.  This  adjustment  will  disturb  the  last  two,  and 
the  three  must  be  repeated  in  succession  until  accurately  adjusted. 

3d.  To  make  the  horizontal  axis  of  the  telescope  perpendic- 
ular to  the  vertical  axis  of  the  instrument. 

To  test,  set  up  the  instrument  as  explained  for  the  last  adjust- 
ment, and,  after  leveling,  center  carefully  a  point  at  each  extremity 
of  the  horizontal  cross-hair.  Turn  the  instrument  upon  its  vertical 


114 


PLANE   SURVEYING  107 

axis  and  transit  the  telescope,  bringing  one  extremity  of  the 
horizontal  cross-hair  upon  one  of  the  points  previously  established. 
If  the  other  extremity  coincides  with  the  second  point,  the  axis  of 
the  telescope  is  in  adjustment. 

If  the  axis  is  out  of  adjustment,  the  method  of  procedure  is 
best  illustrated  by  Fig.  75. 

A  A '  is  the  line  covered  between  the  extremities  of  the  hori- 
zontal wire  or  hair  when  the  axis  of  the  telescope  is  in  adjustment. 
If  it  is  not  in  adjustment,  the  wire  will  in  the  first  position  of  the 


telescope  cover  the  line  A  B,  and  in  the  second  position  the  line 
A  B ' .  Therefore  bisect  the  distance  B  B ' ,  and  raise  or  lower  the 
adjustable  end  of  the  telescope  axis  until  the  wire  covers  A  A ' . 
Now  repeat  the  test,  and  the  correction  if  necessary. 

4th.  To  make  the  axis  of  the  telescope  bubble  tube  parallel  to 
the  line  of  collimatioti  of  the  telescope. 

This  adjustment  should  be  tested  and  corrected  by  the  "  peg  " 
method  as  follows:  Select  a  piece  of  comparatively  level  ground, 
drive  a  stake,  "set  up"  the  transit  over  it,  and  carefully  level  by 


..*- 


• 150' A* 150'  

Fig.  76. 

the  plate  levels.  Next  drive  two  stakes  into  the  ground,  one  in 
front  of  the  transit  and  the  other  at  the  same  distance  behind  it. 
In  Fig.  76,  0  is  the  position  of  the  transit,  and  A"  and  B  are  two 
stakes,  each  150  feet  from  C.  D  is  a  fourth  stake  behind  B  and  in 
line  with  it  from  C.  The  transit  being  leveled  by  the  plate  levels, 
bring  the  bubble  of  the  telescope  tube  to  the  center  of  the  tube,  by 
means  of  the  tangent-screw  attached  to  the  horizontal  axis  of  the 
telescope.  Hold  a  level  rod  upon  A,  adjust  the  target  to  the 


115 


108  PLANE  SURVEYING 

horizontal  cross-hair,  and  note  the  reading.  Unclamp  the  lower 
motion,  turn  the  transit  upon  its  vertical  axis,  and  note  the  reading 
of  a  rod  held  upon  B.  The  difference  of  the  readings  of  the  rod 
held  upon  the  two  points  will  give  the  true  difference  of  level,  no 
matter  how  much  the  telescope  level  may  be  out  of  adjustment. 

Now  take  up  the  transit  and  remove  it  to  the  point  D. 
Carefully  level  the  transit  by  the  plate  levels,  and  again  bring  the 
bubble  of  the  telescope  tube  to  its  center.  Hold  the  rod  upon  the 
point  B  and  note  its  reading.  Do  the  same  at  the  point  A,  and 
take  the  difference  of  the  two  readings.  If  the  telescope  level  is 
in  adjustment,  this  difference  will  be  the  same  as  found  when  the 
instrument  was  over  the  point  C.  Otherwise  the  tube  is  out  of 
adjustment  and  will  be  corrected  as  follows: 

Let  x  represent  the  difference  of  level  of  A  and  B  when  the 
transit  is  at  C. 

Let  y  represent  the  difference  of  level  of  A  and  B  when  the 
transit  is  at  D. 

Let  z  represent  the  difference  between  a?  and  y. 

If  y  is  greater  than  a?,  subtract  z  from  the  rod  reading  upon  A 
for  the  transit  at  D,  and  set  the  target  at  this  new  reading.  Re- 
volve the  telescope  upon  its  horizontal  axis,  by  means  of  the 
tangent-screw,  until  the  horizontal  wire  accurately  bisects  the  target. 
Now  clamp  the  telescope  axis,  and  bring  the  bubble  to  the  center 
of  the  tube  by  means  of  the  capstan-headed  screw  at  one  end 
of  the  tube.  Again  hold  the  rod  upon  B,  and  then  upon  A,  and 
take  the  difference  of  their  readings.  If  this  difference  now  agrees 
with  the  true  difference  of  elevation  of  the  two  points,  the  adjust- 
ment is  complete.  Repeat  the  operation  as  often  as  may  be 
necessary. 

If  y  is  less  than  x,  add  z  to  the  rod  reading  upon  A  for  the 
transit  at  D,  and  set  the  target  at  this  new  reading.  Bisect  the 
target  by  the  horizontal  cross-hair  as  before,  clamp  the  horizontal 
axis,  and  bring  the  bubble  to  the  center  of  the  tube.  Test  and 
repeat  as  described  before. 

Some  transits  are  provided  with  an  adjustable  vernier  to  the 
vertical  circle  or  arc,  which  should  read  0°  when  the  telescope  is 
horizontal.  The  former  adjustment  having  been  completed,  the 
vernier  can  then  be  readily  fixed  in  place. 


116 


PLANE  SURVEYING  100 

The  cross-hair  intersection  should  be  in  the  center  of  the  field 
ot  vision  of  the  eye-piece,  and  this  adjustment  may  be  made  by 
means  of  the  capstan-head  screws  attached  to  the  eye-piece  tube. 

To  "set  up"  the  transit.  Lift  the  instrument  out  of  the 
box  by  placing  the  hands  underneath  the  plates.  Avoid  lifting  it 
by  the  telescope  or  the  standards.  In  attaching  it  to  the  tripod  be 
careful  that  the  threads  engage  properly,  and  screw  it  down  firmly. 
Examine  the  tripod  legs,  and  see  that  they  are  properly  attached 
to  the  tripod  head,  neither  too  tight  nor  too  loose.  See  that  the 
tripod  shoes  are  tight,  and,  before  taking  up  the  instrument, 
lightly  clamp  all  the  movable  parts  to  prevent  unnecessary  wear 
and  straining.  Carry  the  instrument  in  the  most  convenient  way, 
taking  care  not  to  hit  it  against  trees,  lamp-posts,  doors,  etc. 

To  center  the  transit  over  a  stake,  rest  one  leg  of  the  tripod 
upon  the  ground,  and,  grasping  the  other  legs,  pull  the  instrument 
in  the  proper  direction  to  cover  the  stake.  Now  attach  the  plumb- 
line,  and  after  bringing  it  to  rest  as  close  to  the  top  of  the  stake 
as  possible,  note  if  the  point  is  directly  over  the  point  in  the 
stake.  If  it  is  not  too  far  off  the  center,  it  may  be  brought 
closer  by  forcing  the  opposite  leg  into  the  ground  or  by  a  further 
spreading  of  the  legs.  After  the  instrument  has  been  approxi- 
mately centered,  it  may  be  accurately  adjusted  by  means  of  the 
shifting  head.  The  operation  of  "setting  up"  is  difficult  of 
description,  and  facility  can  be  attained  only  by  practice.  Avoid 
having  the  plates  too  much  out  of  level,  as  this  will  result  in 
unnecessary  straining  of  the  leveling  screws  and  plates. 

Having  centered  the  instrument  over  the  stake,  level  it  up  by 
the  levels  upon  the  horizontal  plate.  To  do  this,  turn  the 
instrument  upon  its  vertical  axis  until  the  bubble  tubes  are 
parallel  to  a  pair  of  diagonally  opposite  plate-screws.  Now,  as 
you  stand  facing  the  instrument,  grasp  the  screws  between  the 
thumb  and  forefinger,  and  turn  the  thumb  of  the  left  hand  in  the 
direction  the  bubble  must  move.  Turn  both  thumbs  in,  or  both 
thumbs  out.  Adjusting  one  tube  will  disturb  the  other,  but  adjust 
each  alternately  until  the  bubble  of  each  remains  in  the  center. 

To  measure  a  horizontal  angle  by  means  of  the  transit. 
Set  up  the  transit  over  the  point  C,  Fig.  77.  Set  the  verniers 


117 


110  PLANE  SURVEYING 

to  read  0°,  and  clamp  the  upper  limb.  Now  revolve  the  transit 
upon  its  vertical  axis -by  the  lower  motion,  and  sight  to  A.  Clamp 
the  lower  motion,  and  accurately  adjust  the  intersection  of  the 
cross-hairs  to  the  point  by  means  of  the 
lower  tangent-screw.  Now  unclamp  the  upper 
limb,  and  turn  it  upon  its  upper  vertical 
axis  to  the  point  B.  Clamp  the  upper  limb 
and  adjust  the  line  of  sight  by  the  upper 
tangent-screw.  The  angle  will  now  be  found 
recorded  upon  the  horizontal  circle.  If  it  becomes  necessary  to 
repeat  the  angle,  loosen  the  lower  motion,  and,  without  disturbing 
the  upper  plate,  turn  the  instrument  back  to  the  point  A.  Next 
clamp  the  lower  motion,  loosen  the  upper,  arid  turn  the  telescope 
to  the  point  B.  The  sum  of  the  two  measured  angles  will  now  be 
found  recorded  upon  the  horizontal  circle.  Repeat  as  often  as 
necessary,  and  divide  the  total  horizontal  angle  by  the  number  of 
repetitions  for  the  probable  value  of  the  angle  ABC. 

The  operation  of  laying  off  a  certain  angle  is  essentially  the 
same  as  the  preceding,  except  that  after  the  point  A  has  been 
centered  and  the  required  angle  laid  off  upon  the  horizontal  circle, 
the  tack  in  the  stake  B  must  be  moved  back  and  forth  until  it  is 
accurately  centered  at  the  intersection  of  the  cross-hairs. 

To  survey  a  series  of  lines  by  means  of  the  transit.  "Set  up" 
the  transit  over  the  point  A,  Fig.  78,  and  make  the  verniers  read 
0°.  Have  the  north  end  of  the  plate  ahead,  that  is,  in  the 
direction  of  the  survey.  If  a  true  meridian  line  has  previously  been 
laid  out,  the  declination  of  the  A 
needle  may  be  determined  from  it. 
Without  disturbing  the  upper 
plate,  turn  the  transit  upon  its 
lower  motion,  and  center  upon  B. 
Let  the  needle  down  upon  its 
pivot,  and  as  soon  as  it  has  come 
to  rest  take  the  bearing  of  the  line  FiS-  78' 

AB.  To  measure  the  length  of  the  line,  hold  one  end  of  the  tape  or 
chain  directly  under  the  point  of  the  plumb-bob,  and  send  the  head 
man  in  the  direction  of  B.  As  he  reaches  the  end  of  his  tape, 
place  him  accurately  in  line  by  the  vertical  hair.  For  this  purpose 


118 


PLANE  SURVEYING 


111 


the  telescope  should  be  turned  in  a  vertical  plane  to  bring  the 
intersection  as  closely  as  possible  to  the  top  of  the  stake.  Now 
repeat  this  operation  until  the  point  B  is  reached.  Move  the 
transit  to  the  point  B,  arid  set  it  up  as  before  with  the  north  end  of 
the  plate  ahead.  Transit  the  telescope,  and  examine  the  verniers 
to  see  if  they  read  0°.  By  means  of  the  lower  motion,  center  upon 
A,  lower  the  needle,  and  take  the  bearing.  Now  clamp  the  lower 
motion,  transit  the  telescope,  and  revolve  the  upper  plate  so  the 
telescope  points  to  C.  Read  the  angle,  which  is  that  between  A  B 
produced  and  B  C.  Also  read  the  bearing  from  the  compass.  Now 
measure  B  C  as  explained  before,  take  up  the  instrument,  and  set 
it  up  at  the  point  C.  Continue  thus  until  the  series  of  lines  have 
been  surveyed. 

The  angles  measured  are  those  indicated  by  the  dotted  lines 
(Fig.  78),  and  are  called  deflection  angles. 

It  is  desirable  in  work  of  this  character  to  calculate  the  bearing 
or  azimuth  of  a  line  from  the  deflection  angle,  and  to  check  the 
result  by  the  compass.  Referring  to  the  figure,  the  records  of  the 
survey  will  be  kept  as  follows,  using  the  left-hand  page  of  the 
transit  or  field  book  and  starting  from  the  bottom  of  the  page: 


DEFLECTION.                         ;      DKJ)UCKn 

LEFT. 

RIGHT. 

BEARINGS. 

7  +  95 

275 

52°   0' 

852°  45'  E 

S52°30'E 

5  +  30 

265 

51°  0' 

N75°15'  E 

N75°30'  E 

2  +  90 

240 

31°  30' 

8  53°  45'  E 

8  53°  30'  E 

0                  290 

S85°E 

It  is  not  absolutely  necessary  An  the  method  described  above, 
that  the  verniers  should  be  set  to  read  0°  before  aligning  the 
instrument,  provided  that  the  verniers  are  read  before  turning  off 
the  angle.  For  instance,  if  the  verniers  read  40°  15'  after  the 
instrument  is  set  up  over  a  stake,  and  after  sighting  along  a  certain 
line,  read  60°  15',  the  angle  between  the  two  lines  is  the  difference 
of  the  reading  of  the  verniers,  or  20°. 

As  already  explained  under  "the  compass,"  the  direction  of  a 
line  may  be  given  by  its  azimuth.  If  azimuths  are  computed  from 


110 


112 


PLANE   SURVEYING 


the  north  to  the  right  through  360°,  the  azimuths  for  the  preceding 
case  will  be  as  follows,  as  illustrated  by  the  diagram : 

Since  the  bearing  of  the  first  line  A  B  (Fig.  79),  given  by 
the  compass,  is  S  85°  E,  its  azimuth  will  evidently  be  the  difference 
between  180°  and  85°,  or  95°.  Since  the  second  line  B  C 
deflects  to  the  right  by  31°  30',  its  azimuth  will  evidently  be  the 
sum  of  the  angles  95°  and  31°  30',  or  126°  30'.  Since  the  third 
line  C  D  deflects  to  the  left  51°  0',  its  azimuth  will  be  the  differ- 
ence of  the  angles  126°  30'  and  51°  0',  or  75°  30'.  The  fourth 
line  D  E  deflects  to  the  right  52°  0',  arid  there  fore  its  azimuth  will 
\.e  the  sum  of  the  angles  75°  30'  and  52°  0',  =  127°  30'. 


The  same  diagram  may  serve  to  illustrate  the  method  of 
deducing  the  bearings  of  a  series  of  lines  from  the  deflection 
angles.  Since  the  bearing  of  the  first  line  as  given  by  the  compass 
is  S  85°  E,  and  the  second  line  deflects  to  the  right  31°  30',  it  is 
evident  that  this  second  line  decreases  its  easting  by  that  amount, 
so  that  its  bearing  will  be  the  difference  between  85°  and  31°  30', 
=  S  53°  30'  E.  Since  the  third  course  deflects  to  the  left  by  51° 
0',  its  bearing  to  the  east  will  be  increased  by  this  amount,  but  will 
pass  into  the  northeast  quadrant  by  14°  30',  making  its  bearing 
90°— 14°  30',  =  N  75°  30'  E.  The  fourth  course  deflects  to  the 
right  52°  0',  returning  to  the  southeast  quadrant  by  37°  30', 
making  its  bearing  90°— 37°  30',  or  S  52°  30'  E. 


120 


PLANE   SURVEYING 


Traversing.  This  is  a  method  of  observing  and  recording  the 
directions  of  a  series  of  lines  of  a  survey,  so  as  to  read  off,  upon 
the  horizontal  circle,  the  angles  that  the  lines  make  with  some 
other  line  of  the  survey,  which  may  be  either  a  true  meridian  or 
some  line  adopted  as  a  meridian  for  that  survey. 

Before  starting  out  upon  a  traverse,  it  is  best  to  lay  out  upon 
the  ground  a  true  meridian,  either  from  observations  on  Polaris  or 
by  means  of  the  "Solar  transit,"  as  will  be  explained  later  on.  This 
line  should  be  300  or  400  feet  in  length,  clearly  denned  by  stakes 
carefully  centered,  one  of  the  stakes  preferably  being  the  first 
station  of  the  survey.  The  transit  can  now  be  set  over  this  stake, 
and  the  line  of  sight  carefully  centered  upon  the  second  stake  by 
means  of  the  lower  motion,  the  verniers  first  having  been  set  to 
read  0°.  The  subsequent  operations  can  best  be  illustrated  by  a 
diagram. 


Fig.  80. 

First  lay  out  a  meridian  through  station  1  (Fig.  83),  and 
define  it  by  a  stake  driven  300  or  400  feet  away  towards  N.  Both 
of  these  stakes  should  be  carefully  "  witnessed,"  that  they  may  be 
recovered  at  any  time.  To  begin  the  survey,  carefully  center  the 
transit  over  station  1,  with  verniers  set  to  zero;  turn  the  instrument 
upon  its  lower  motion  until  the  line  of  sight  approximately 
covers  the  stake  at  the  north  end  of  the  meridian,  and  carefully 
center  it  by  the  lower  tangent-screw;  lower  the  compass  needle  (if 
there  is  one),  and  note  and  record  the  magnetic  declination.  Next, 
with  the  lower  motion  securely  clamped,  unclamp  the  upper  motion, 
and  revolve  the  upper  plate  in  the  direction  of  station  2.  Clamp 
the  upper  motion,  and  carefully  center  the  line  of  sight  by  the 


121 


114  PLANE  SURVEYING 

upper  tangent-screw.  Note  and  record  the  angle  upon  the  plate, 
which  will  be  the  azimuth  of  the  line  1  2.  Measure  the  distance 
from  1  to  2,  and  note  and  record  the  compass  bearing  as  a  check. 

Now,  with  the  upper  motion  securely  clamped,  remove  the 
transit  to  station  2;  and  carefully  center  it  over  the  stake  with  the 
north  end  of  the  plate  ahead,  that  is,  in  the  direction  from  1  to  2. 
Transit  the  telescope,  unclamp  the  lower  motion,  and  bring  the 
line  of  sight  to  cover  station  1.  Carefully  center  it  by  the  lower 
tangent-screw.  Clamp  the  lower  motion,  transit  the  telescope, 
unclamp  the  upper  motion,  and  revolve  the  upper  plate  until  the 
line  of  sight  falls  upon  station  3,  carefully  centering  it  by  the  upper 
tangent-screw.  Read  and  record  the  plate  angle,  which  will  be  the 
azimuth  of  the  line  2  3.  Measure  the  distance  from  2  to  3,  and  read 
the  bearing  of  the  needle  for  a  check. 

Now  see  that  the  upper  plate  is  securely  clamped,  move  the 
instrument  to  station  3,  and  proceed  as  before;  and  so  on  through- 
out the  traverse. 

In  the  above  example  (Fig.  80),  since  the  first  line  is  in  the  north- 
east quadrant,  its  bearing  N.  38°  E.  will  be  the  same  as  its  azimuth 
(38°),  and  this  angle  is  recorded  upon  the  plate.  The  second  line, 
however,  passed  into  the  southeast  quadrant,  and  its  azimuth  will 
be  recorded  upon  the  plate  as  38°  +  81°,  or  119°.  Its  bearing,  how- 
ever, will  be  S.  61°  E.  The  azimuth  of  the  third  line  will  be  38°  + 
81°  -  23°,  or  96°.  Its  bearing  will  be  S.  84°  E.  In  a  similar 
manner,  the  bearing  of  the  fourth  line  may  be  deduced  from  the 
azimuth  as  shown  upon  the  plate. 

Fig.  81  illustrates  a  traverse  beginning  in  the  southeast  quadrant. 
The  bearing  of  the  first  line  is  S.  41°  E.,  and  therefore  its  azimuth 
will  be  180°  -  41°  =  139°.  The  bearing  of  the  second  line  will  be 
S.  73°  E.,  and  its  azimuth  will  be  107°;  and  so  on  with  the  remaining 
courses  of  the  traverse. 

By  a  similar  process  of  reasoning,  the  azimuths  of  courses  in 
the  southwest  and  northwest  quadrants  may  be  deduced  from  the 
bearings,  and  vice  versa. 

We  therefore  establish  the  following  rules: 

(a)     Courses  in  northeast  quadrant — Azimuth  =  Bearing. 
(6)     Courses  in  southeast  quadrant — Azimuth  =  Supplement  of  Bear- 
ing. 


122 


PLANE  SURVEYING  115 


(c)  Courses  in  southwest  quadrant — Azimuth  =  180°  +  Bearing. 

(d)  Courses  in  northwest  quadrant— Azimuth  =  360°  -  Bearing. 

Bearing  north:    Azimuth  =  0°  or  360°. 
"        east:  "         =  90°. 

"        south:  "         ==  180°. 

"         west:  "         =  270°. 

For  convenience  in  determining  azimuths,  the  inside  graduations 
of  the  horizontal  circle  may  be  numbered  from  0°  to  360°  to  the 


Fig.  81. 

right,  beginning  at  the  north  end;  and  the  outside  graduations  from 
0°  to  360°  to  the  right,  beginning  at  the  south  end.  Then  keeping 
the  north  end  of  the  plate  ahead — that  is,  in  the  direction  of  the 
traverse — for  a  traverse  beginning  in  either  the  northeast  or  north- 
west quadrants,  azimuths  will  be  recorded  directly  upon  the  plate 
from  the  inside  graduations;  and  for  a  traverse  beginning  in  either  the 
southeast  or  southwest  quadrants,  azimuths  will  be  recorded  directly 
upon  the  plate  from  the  outside  graduations. 

Traversing  is  particularly  adapted  to  surveying  roads,  streets, 
railroads,  shores  of  lakes,  river  banks,  etc.;  and  in  land  surveying  it 
possesses  an  advantage  over  the  method  by  interior  angles,  on  account 
of  the  readiness  it  affords  in  obtaining  the  bearings  from  the  azimuths, 
and  the  greater  rapidity  with  which  the  work  may  be  platted,  since 
the  angle  which  each  line  makes  with  the  assumed  meridian  or  refer- 
ence line,  may  be  taken  at  once  from  the  field  notes. 

In  United  States  government  surveys,  when  a  traverse  is  run  to 
mark  the  divisions  between  private  estates  and  bodies  of  water 
retained  as  public  property,  it  is  called  a  meander  line. 


110 


PLANE  SURVEY  ING 


Keeping  Notes.  All  notes  of  measurements  of  angles  and  dis- 
tances should  be  recorded  as  soon  as  made,  in  a  special  notebook 
adapted  to  the  purpose.  Avoid  the  practice  of  making  notes  upon 
scraps  of  paper  and  in  small  pocket  notebooks  and  of  filling  in  details 
from  memory.  The  notes  will  probably  Jbe  used  by  other  persons 
unfamiliar  with  the  locality,  for  platting  and  for  general  information, 
and  these  persons  must  depend  entirely  upon  what  is  recorded,  and 
how  it  is  recorded,  for  their  interpretation.  To  this  end  the  notes 
should  be  clear  and  concise,  yet  full  enough  to  give  all  necessary 
information.  They  should  permit  of  only  one  interpretation,  and 
that  the  correct  one. 

The  note  keeper  should  bear  in  mind  constantly  the  nature  of 
the  survey  and  the  object  to  be  attained,  and  this  will  enable  him  to 
determine  what  measurements  are  necessary.  Do  not  crowd  the 
notes.  Use  the  left-hand  page  of  the  book  for  notes,  and  the  right- 
hand  page  for  such  sketches  and  remarks  as  may  be  necessary.  The 
record  is  usually  made  with  a  pencil,  using  a  medium  hard  grade. 
If  incorrect  entries  are  made,  erase  them  neatly;  but  avoid  errors  as 
much  as  possible,  as  too  many  erasures  tend  to  discredit  the  work. 
After  each  notebook  is  filled,  label  it  with  the  subject  of  the  survey, 
the  dates  between  which  it  was  recorded,  and  all  other  information 
that  may  be  of  service  in  filing  for  future  reference.  Above  all,  do 
not  lose  a  notebook,  as  it  may  contain  information  that  cannot  be 
recovered  at  any  price. 

Referring  to  Fig.  80,  the  notes  would  be  entered  in  the  record 
as  follows: 


STATION 

DISTANCES 

DEFLECTIONS 

AZIMUTH 

NEEDLE 

RIGHT 

LEFT 

1 
2 
3 
4 

220  .  00' 
225.00' 
235  .  00' 
190.00' 

38°  00' 
81°  00' 

23°  00' 
40°  00' 

38°  00' 
11  9°  00' 
96°  00' 
56°  00' 

N.  38°  00'  E. 
S.  61°  00'  E. 
S.  84°  00'  E. 
N.  56°  00'  E. 

Figs.  82  and  83  illustrate  the  method  of  keeping  notes  by  means 
of  sketches. 

Checking  the  Traverse.  For  an  ordinary  survey  not  involving 
unusual  precision,  a  transit  reading  to  single  minutes  will  be  sufficient. 
A  single  measurement  will  ordinarily  give  the  angle  with  sufficient 


PLANE  SURVEYING 


117 


accuracy;  but  should  a  check  upon  the  measurement  be  deemed 
necessary  the  angle  may  be  "repeated"  as  explained  on  page  109. 

As  a  further  check  against  errors  in  angles,  the  magnetic  bearing 
of  each  line  should  be  read,  showing  the  approximate  directions  of 
the  lines,  and  by  comparison  with  the  azimuths  or  the  deduced  bear- 
^,  ings,  will  serve  to  point  out  gross 
errors,  as  for  instance  reading  right 
for  left  when  measuring  deflec- 
tion angles.  This  check  upon  the 


Fig.  82. 


Fig.  88. 


angles  should  always  be  applied  in  the  field,  so  that  errors  may  be 
rectified  before  leaving  the  work. 

If  the  traverse  involves  a  closed  area,  the  accuracy  of  the  transit 
work  may  be  tested  by  adding  together  all  of  the  measured  angles. 
The  sum  of  the  interior  angles  should  equal  (n-2)  X  180°,  n  being  the 
number  of  sides  of  the  field.  For  the  deflection  angles,  the  sum  of 
all  the  deflections  to  the  right  should  differ  from  the  sum  of  all  the 
deflections  to  the  left  by  360°;  that  is  to  say,  the  algebraic  sum  of  the 
deflection  angles  should  be  360°. 

It  is  sometimes  desirable  to  check  the  lengths  of  the  courses 
before  leaving  the  field.  If  there  is  any  reason  to  question  the  accuracy 


PLANE  SURVEYING 


of  the  measurement  of  any  line,  it  should  be  measured,  preferably  in 
the  opposite  direction.  In  a  closed  traverse,  it  is  well  to  run  diagonal 
lines  across  the  traverse  as  an  additional  check  upon  both  the  angles 
and  the  measured  distances. 

For  city  work,  the  engineer  should  lay  out  a  true  meridian 
300  or  400  feet  long,  and  mark  the  extremities  of  the  line  by  per- 
manent monuments  set  in  the  ground  and  carefully  protected 
from  disturbance.  To  do  this,  in  some  convenient  place  permit- 
ting an  unobstructed  line  of  eight,  drive  a  large  stake,  and  mark 

its  center  by  a  hollow -headed  tack. 
Center  the  transit  carefully  over 
this  point,  and  proceed  to  lay  out 
a  true  meridian,  preferably  with 
the  solar  attachment.  Mark  the 
direction  of  this  line  by  a  second 
stake  carefully  centered  by  a  tack 
as  before.  Now,  about  25  feet 
from  the  first  stake,  and  in  line 
with  it  and  the  second  stake,  exca- 
vate a  hole  in  the  ground  about 
three  feet  in  diameter  and  five  feet 


Fig.  84. 


deep,  or  deep  enough  to  be  below  the  frost  line.  Next  build  a 
foundation  of  concrete  about  two  feet  square  and  three  feet  deep. 
Before  this  has  "  set,"  insert  in  it  a  cut-stone  post  about  nine 
inches  square  at  its  lower  end  and  of  such  length  that  its  top  will 
come  just  below  the  surface  of  the  ground,  and  having  set  into  its 
top  a  copper  bolt  about  £  inch  by  4  inches.  The  post  may  be 
centered  in  the  concrete  by  the  transit,  and  should  be  set  "plumb." 
Now  locate  and  build  a  second  monument  in  line  with  the 
second  stake  and  a  few  feet  from  it.  After  the  concrete  has  set 
firmly,  again  set  the  transit  carefully  over  the  center  of  the  first 
stake,  and  accurately  align  it  by  the  tack  in  the  second  stake.  Now 
"plunge"  (reverse)  the  telescope,  and  carefully  center  a  point  in 
the  top  of  the  copper  bolt;  mark  this  point  with  a  steel  punch.  In 
the  same  way  center  a  point  in  the  top  of  the  bolt  of  the  second 
monument.  The  monuments  may  be  protected  by  enclosing  them 
in  cast-iron  valve-boxes  with  covers.  Either  one  or  both  of  these 
monuments  may  be  used  as  "standard"  bench-marks  from  which 


PLANE  SURVEYING 


111) 


all  the  levels  and  grades  may  be  ascertained.  For  this  purpose  a 
city  datum  may  be  assumed,  or  better,  the  bench-mark  may  be 
connected  by  a  line  of  levels  with  a  bench-mark  of  the  II.  S.  Coast 
and  Geodetic  Survey,  or,  if  such  is  not  available,  with  a  bench- 
mark of  the  nearest  railroad. 

THE  STADIA. 

Attached  to  the  diaphragm  carrying  the  horizontal  and  vertical 
hairs,  are  two  auxiliary  horizontal  hairs  called  "stadia''  hairs  or 
wires.  These  hairs  may  be  either  fixed  in  position  or  adjustable, 


but  the  fixed  hairs  are  the  better  for  field  use  and  cost  much  less. 
See  Fig.  84.  Any  instrument-maker  will  equip  a  level  or  a 
transit  with  either  fixed  or  adjustable  stadia  wires,  and  they  should 
be  included  in  every  outfit. 

The  stadia  is  used  for  measuring  horizontal  distances  and 
differences  of  elevation,  without  the  use  of  chain  or  tape  or  other 
apparatus  except  the  leveling  rod  or  a  specially  graduated  stadia 


Fig.  86. 

rod.  It  is  based  upon  the  principle  of  the  similarity  of  triangles. 
Thus,  if  the  stadia  hairs  are  spaced  so  as  to  intercept  one  foot  upon 
a  rod  held  at  a  distance  of  one  hundred  foot,  the  rod  intercept 
for  any  other  distance  will  be  in  direct  proportion  to  the  first. 
See  Fig.  85. 


120 


PLANE  SURVEYING 


Unfortunately  the  construction  of  the  telescope  of  an  engineer- 
ing instrument  modifies  the  above  simple  statement,  and  a  formula 
for  the  use  of  the  stadia  will  now  be  deduced. 

Let  O  in  Fig.  86  be  the  optical  center  of  the  object-glass  of 
the  telescope.  This  point  may  be  assumed  at  the  center  of  the 
lens,  and  the  error  involved  in  such  assumption  is  inappreciable 
and  may  be  neglected. 

Let  SSi  be  a  portion  of  the  stadia  rod  covered  by  the  stadia 
hairs  CCi.  From  C  and  Ci  draw  the  lines  CSi  and  CiS  through  the 
optical  center  of  the  object-glass.  Upon  looking  through  the  eye- 
piece of  the  telescope,  O  will  be  seen  a^  at  Si  and  Ci  as  at  S. 


C3c, 


Fig.  87. 


Call  i  the  distance  between  the  stadia  hairs,  s  the  intercept 
upon  the  rod,/'  the  distance  from  O  to  the  wires,  and  d  the 
distance  of  the  rod  from  O. 

The  triangles  COCi  and  SOSi  are  similar,  and  therefore  we  have 
the  proportion 

i:s  ::/'    :  a 


therefore  d  = 


(1) 


But/'  varies  with  d.  That  is  to  say,  if  the  rod  were  to  be 
moved  closer  to  the  instrument,  as  at  QJ),  the  lens  would  be  moved 
farther  from  the  wires,  or  the  wires  from  the  lens,  and  in  either 
case  the  wire  interval  would  intercept  a  shorter  space  upon  the  rod, 

as  S2S3.     The  ratio  -*C— -,  or  its    equal   — ,  will    therefore    vary 

I1  S 

for  each  position  of  the  rod.     But  however  they  vary,  we  have  from 
a  well-known  principle  of  optics: 


PLANE  SURVEYING  121 


+- 

in  which  f  is  the  principal  focal  distance  of  the  lens,  and  f '  and 
d  are  any  pair  of  conjugate  focal  distances.  Substituting  the 

value  of  - .  7  from  (1)  in  (2),  there  results  the  equation 
d=ts+f.  (3) 

Equation  3  gives  the  distance  of  the  rod  from  the  lens. 
We  can  establish  some  very  important  relations: 
In  Fig.  87  lay  off  OF '  =  OF  =  principal  focal  distance  of 

lens  =f. 

C  and  Ci  being  the  stadia  wires,  draw  C  D  and  Ci  E  parallel  to 

the  axis  of  the  lens,  and  through  F'  draw  D  Si  and  E  S;  then  will 

S  Si  =  the  intercept  upon  the  rod. 

The  distance  of  the  rod  from  the  point  F'  is 

<T  =<*-/=(-{-*+/)-/ =-4. 

From  the  similarity  of  the  triangles  EF'O,  SF'B  and  SaF'B',  we 
have: 

F'O  :  OE  ::  F'B  :  BS  ::  F'B'  :  B'S2. 

Therefore  the  points  S,  82,  F',  and  E  lie  in  the  same  straight 
line;  and  therefore,  wherever  the  cross-hairs  are  situated,  or,  more 
strictly,  whatever  may  be  the  position  of  the  lens,  the  visual  lines 
denned  by  the  stadia  wires  will  intercept  the  elements  of  the  cone 
of  light  denned  by  SF'Si. 

All  distances   must  be   measured  from   the    center    of    the 

instrument;   and  therefore  to  the  expression  d  =-' .  s  -{- f  must  be 

added  a  quantity  that  will  represent  the  distance  from  the  center 
of  the  lens  to  the  plumb-line.  This  quantity  is  variable,  of  course, 
but  an  average  value  is  usually  taken.  Call  this  quantity  c.  The 
quantity  /may  be  found  by  focusing  the  instrument  on  a  star,  and 
measuring  the  distance  from  the  center  of  the  lens  to  the  cross- 
hairs. We  can  therefore  determine  the  quantity /+  c.  This, 
however,  is  usually  supplied  by  the  instrument-maker,  and  with 
greater  precision. 


122  PLANE   SURVEYING 

The   complete  equation,  therefore,  for  any  distance  measured 
with  the  stadia,  is: 

+  (/+<?)  (4) 


If,  then,  upon  level  ground  we  lay  off  the  distance  f  +  c  hi 
front  of  the  plumb-line,  drive  a  stake,  measure  from  this  stake  a 
distance  of  100  or  200  feet,  or  any  other  convenient  distance,  and 

f 
note  the  rod  intercept,  then  in  the  formula^'  =   .  s,  d'  and  6'  are 

/'     * 
measured,  and  we  can  determine  the  ratio   .  ;  or,  jf  f  has  been 

previously  determined,  wre  can  determine  the  value  *  or  the  distance 
between  the  cross-hairs. 

f 
The  ratio     —  being   known,    distances    can   be    found    from 

equation  4.  It  is  usually  most  convenient  to  make  this  ratio  100, 
so  that  at  a  distance  of  100  feet  the  wires  will  intercept  one  foot 
upon  the  rod. 

The  rod  may  be  either  an  ordinary  leveling  rod,  or  a  stadia 
rod  divided  specially  for  the  telescope  and  wires.  When  the  rod  is 
specially  graduated,  it  may  be  in  either  one  of  two  ways.  Either 
it  may  be  graduated  so  as  to  give  the  distance  from  F',  in  which 
case  the  quantity  f  +  c  will  have  to  be  added  in  each  instance  ; 
or  it  may  be  graduated  to  give  distances  from  the  center  directly. 

If  the  rod  is  to  be  graduated  specially,  proceed  as  follows  : 

Carefully  level  the  line  of  collimation  of  the  telescope,  and  lay 
off  from  the  plumb-line  the  distance  f  -{-  c.  From  the  point 
thus  established  measure  off  any  convenient  distance,  as  500  feet, 
on  a  horizontal  plane.  Set  up  the  rod,  not  yet  graduated,  at  this 
point,  and  hold  it  carefully  perpendicular  to  the  line  of  sight  from 
the  telescope.  This  can  best  be  done  by  means  of  a  plumb-line. 
Be  careful  to  eliminate  all  parallactic  motion  of  the  wires  on  the 
rod,  when  the  eye  is  moved  up  and  down  before  the  eye-piece. 

Mark  on  the  rod  very  carefully  the  apparent  place  of  the  lower 
wire.  This  should  be  about  one-quarter  the  length  of  the  rod  from 
one  end  if  the  horizontal  distance  first  laid  off  is  about  one-half 
the  greatest  distance  for  which  the  rod  can  be  used.  The  middle 
wire  will  then  be  at  about  the  middle  of  the  rod,  and  the  upper 
one  at  about  one-quarter  the  length  of  the  rod  from  the  other  end. 


130 


PLANE   SURVEYING 


Mark  the  latter  point  carefully.  The  wire  interval  for  a  space  of 
500  feet  from  F'  has  thus  been  found.  One-fifth  of  this  space  will 
be  the  wire  intercept  at  a  distance  of  100  feet;  twice  the  space, 
the  intercept  for  1,000  feet;  and  so  on.  The  intermediate  spaces 
can  thus  be  graduated.  It  must  not  be  forgotten  that  in  using 
the  rod  thus  graduated,  the  quantity  f  -\-  c  nmst  be  added  to  the 
distance  indicated  by  the  rod,  to  reduce  the  distance  to  the  center 
of  the  instrument. 

If  the  rod  is  to  be  graduated  to  give  distances  from  the  center 
of  the  instrument  directly,  proceed  as  before,  marking  the  spaces 
upon  the  rod  corresponding  to  the  distances  measured  upon  the 
ground.  The  quantity  f  -\-  c  will  not  now  have  to  be  added 


to  the  distances  given  by  the  rod;  but  for  every  point  other  than 
that  for  which  the  rod  is  graduated,  the  distance  will  be  in  error 
by  some  fractional  part  of/*-}-  c.  The  reason  for  this  will  be 
apparent  by  referring  to  Fig.  87.  If  the  distance  is  less  than 
that  for  which  the  rod  was  graduated,  the  rod  readings  will  indicate 
too  small  a  distance;  and  for  a  distance  greater  than  the  standard, 
the  rod  readings  will  indicate  a  distance  too  great.  It  is  therefore 
more  exact  to  mark  the  wire  interval  at  100  feet,  200  feet,  and  so  on 
through  the  length  of  the  rod.  Each  space  thus  determined  can 
be  divided  up  as  desired;  and  the  error  involved  in  any.  reading 
will  then  be  much  smaller  than  if  the  rod  were  graduated  for  a 
single  standard  distance  only. 


124  PLANE   SURVEYING 

Thus  far  the  rod  has  been  assumed  as  held  perpendicular  to 
the  line  of  sight,  which  of  course  will  always  be  the  case  when 
using  the  stadia  in  the  leveling  instrument.  The  stadia,  however, 
finds  its  greatest  usefulness  in  connection  with  the  transit,  when 
the  line  of  sight  is  seldom  horizontal.  If,  at  the  same  time  the 
rod  intercept  is  read,  the  vertical  angle  is  noted,  differences  of 
elevation  may  be  determined,  as  well  as  the  distances. 

A  formula  will  now  be  deduced  for  reducing  inclined  readings 
to  the  horizontal,  and  for  determining  differences  of  elevation,  the 
rod  being  held  vertical. 

In  Fig.  88,  let  the  angle  of  inclination  of  the  line  of  sight  to 
the  horizontal  plane  be  called  BCN  =  FBD  =  I.  This  angle 
will  be  measured  upon  the  vertical  circle  of  the  transit.  If  the 
rod  be  held  perpendicular  to  the  line  of  sight,  the  intercept  upon 
the  rod  =  D  E  =  ,v.  Represent  the  rod  intercept  when  the  rod  is 
held  vertical  by  $ ' .  Now  since  the  angle  F  D  B  =  90°  nearly, 

D  E  =  F  G  cos  I,  or  s  =  s '  cos  I.    But  C  B  =  4-«  +  (/+<?)  = 


cos  I  +  (f  -{-  c}.    Therefore  the  horizontal  distance  to  the  rod 


=  C  B  cos  I  =  •£* '  cos2 1  +  (/  +  c)  cos  I  =  C  N.     The  vertical 

distance  of  the  point  B  above  the  horizontal  plane  through  the 

f 
axis  of  the  telescope  =  B  N  =  C  B  sin  I  =  - - .  s'  cos  I  sin  I  + 

i 

(/+  c}  sin  I  -  |  -•£*'  sin  21  +  (/+  c)  sin  I. 

% 

For  vertical  angles  less  than  5°  the  quantity  (f  +  c)  sin  I  is 
less  than  0.1  (f  +  c)  and  may  be  neglected. 

The  Use  of  the  Stadia  in  the  Field.  In  using  the  stadia  wires 
in  level  country,  no  special  instructions  are  necessary,  as  the  line  of 
sight  is  at  all  times  horizontal.  Over  very  uneven  ground,  the 
use  of  the  level  and  stadia  is  very  limited.  However,  there  are 
often  conditions  in  which  the  stadia  wires  in  a  leveling  instrument 
are  a  very  great  convenience.  The  range  of  the  instrument  may 
sometimes  be  increased  by  using  the  center  wire  together  with  one 
of  the  stadia  wires,  but  the  instrument  should  be  carefully  tested 
to  ascertain  if  the  stadia  wires  are  equally  spaced  with  reference  to 
the  middle  wire. 


PLANE  SUKVEYING 


For  extended  surveys  over  uneven  country,  the  transit  and 
stadia  are  particularly  adapted,  and  especially  for  filling  in  details 
of  an  extended  topographical  survey.  The  saving  of  time  and 
expense  are  important  elements  in  favor  of  the  transit  and  stadia 
as  compared  with  the  transit  and  tape ;  and  with  a  little  practice 
and  attention  to  details  the  results  should  be  fully  as  accurate. 
Certainly,  when  an  engineer  must  depend  upon  unskilled  help  to 
carry  the  tape,  there  can  be  no  choice  as  to  which  to  use. 

For  use  with  the  stadia,  the  transit  should  be  provided  with  a 
complete  vertical  circle,  reading  to  minutes  at  least ;  and  a  level 
tube  should  be  attached  to  the  telescope.  The  eye-piece  should  be 
inverting.  Before  starting  out  upon  a  survey,  the  transit  should 
be  carefully  tested  and  corrected  through  all  of  its  adjustments. 
The  field  operations  are  then  as  follows : 

Set  up  the  instrument  over  a  principal  station  of  the  survey, 
and  level  it  carefully.     If  a  solar  attachment  is  available,  it  will  be 
desirable  to  lay  out  a  true  meridian,  from  which  the  declination  of 
^s^^s^^^f      the   needle    may    be    determined. 
Now  determine  the  height  of  the 
cross-hairs   by  holding  the  stadia 
rod    close    to     the     side    of    the 
instrument,  and  noting  the  height 
of    the    center  of   the    horizontal 
axis     of     the    telescope.      Locate 
Fig.  89.  the  second   station   carefully,  and 

turn  the  telescope  upon  the  horizontal  axis  until  the  center  wire 
cuts  the  division  upon  the  rod  (held  upon  the  ground)  representing 
the  height  of  the  axis  of  the  instrument  above  the  ground  at  the 
first  station.  Now  determine  the  azimuth  of  the  line  connecting 
the  two  stations,  read  the  vertical  angle  of  the  telescope,  and 
determine  the  rod  intercept.  Enter  these  items  in  the  field  book 
and  proceed  to  take  observations  upon  sub-stations  (called  "side- 
shots"). 

The  same  program  is  to  be  repeated  for  each  case,  except  that 
the  side-shots  may  or  may  not  be  taken  upon  points  indicated  by 
stakes.  The  principal  stations  of  a  stadia  survey  should  be 
ixjrmaneiit:  the  stakes  should  be  driven,  and  "witnessed"  so  as  to 
be  easily  recovered. 


126 


PLANE   SURVEYING 


Having  now  located  all  the  necessary  whits  from  the  first 
station,  remove  the  instrument  to  the  second  station,  arid  set  it  up 
with  the  north  end  of  the  plate  in  the  direction  of  the  survey. 
Having  carefully  leveled  the  instrument,  determine  the  height 
of  its  axis  as  before,  and  send  the  rod  back  to  the  first  station. 
Transit  the  telescope,  and  sight  upon  the  rod  as  before.  Read 
vertical  angle  and  stadia  rod,  and  determine  azimuth,  and  these 
will  serve  to  check  the  former  determinations. 

In  moving  from  one  station  to  another  it  is  advisable  to  set 
the  scale  of  the  horizontal  circle  to  zero.  Transit  the  telescope 
again  and  locate  the  next  station;  and  so  on  throughout  the  survey. 

The  principal  stations  of  a  stadia  survey  may  have  been 
located  by  a  previous  triangulation.  in  which  case  it  will  probably 
be  necessary  to  locate  intermediate  stations  as  the  survey 
progresses.  Or  all  of  the  stations  may  be  located  during  the 
progress  of  the  survey.  The  courses  connecting  the  principal 
stations  form  the  "backbone"  of  the  survey,  and  the  azimuths 
and  distances  should  be  checked  at  every  opportunity. 

In  keeping  the  field  notes,  represent  the  principal  stations  by 
triangles,  as  Ai  A2  As,  etc.;  and  the  secondary  stations  by  circles, 
as  0i  ©2  ©3,  etc. 

Below  will  be  found  an  example  of  the  method  of  keeping 
notes.  Use  the  right-hand  page  for  sketches,  or  for  such  additional 
notes  as  may  be  necessary. 


A  0 

H.  i.  5.15 

KLEV.  100. 

VEUT. 
ANGLE. 

DIFFER- 
ENCE. 

ELEV. 

©   1 

238                S 

88°  38'  W 

57' 

—  3.9 

96.1 

©     2 

265                S 

57°  41'  W 

58' 

—     .6 

95.4 

©     3 

236                S 

49"    3'  W 

38' 

-     .6 

95.4 

4 

237                S 

32°  58'  W 

—  r  T 

-     .6 

95.4 

5 

261               S 

0°  13'  W 

—  1°  3' 

-     .6 

95.4 

A    1 

425                S 

44^  13'  E 

—  0°   6' 

—     .5 

99.25 

©     0 

H.  I.  475      S 

135°  41'  E 

6 

245               S 

200°  23'  W 

+  1°57' 

+  8.3 

107.6 

7 

300               S 

204°    2'  W 

+  1°31' 

-}-  7.9 

101.2 

8 

345                S 

203°  12'  W 

+      58' 

-f  5.8 

105.1 

&    2 

Stadia  Rods.    Telemeter  or  stadia  rods  are  made  of  clear  white 
pine  well  seasoned,  about  &  of  an  inch  thick,  from  4  to  4^  inches 


PLANE   SURVEYING 


1-21 


wide,  and  from  10  to  16  feet  long.  They  are  protected  by  a  metal 
shoe  to  keep  the  lower  end  from  being  battered  or  split.  The  rod 
is  stiffened  by  having  a  piece  2|  inches  wide  along  its  back. 
Generally  a  stadia  rod  is  hinged  at  the  center  for  greater  con- 
venience in  transportation,  and  at  the  same  time  it  is  provided 
with  a  bolt  on  the  back  to  protect  the  graduation  and  to  hold  it  in 
position  when  in  use. 

A  self -reading  level  rod  may  be  used  for  distances  if  the  wires 

are  adjustable  (see  Figs.  89  and  84),   or  if  the  wire  interval  has 

A 


Fig.  92. 


Fig.  91. 

been  determined  in  standard  units.  The  rods 
used  in  connection  with  this  grade  of  work  differ 
from  those  employed  in  ordinary  leveling. 
Those  with  graduations  have  the  inner  surface 
recessed  to  protect  the  graduated  surface,  and 
Fig.  90.  are  painted  white  with  the  scale  in  black.  The 
forms  of  graduation  are  different  on  different  rods.  In  some,  the 
unit  of  measure  is  the  meter,  while  others  have  the  foot,  as  will  be 
described  later.  When  telemeters  are  in  use,  they  are  open,  laid 
flat,  and  held  securely  in  line  by  the  brass  clip  (or  bolt)  above 
referred  to.  They  are  sometimes  provided  with  a  target. 

In  order  to  have  the  rod  held  in  a  perfectly  vertical  position, 
a  small  telescope  is  sometimes  attached  to  its  side,  by  means  of 


128  PLANE  SURVEYING 


M 


O 


M 


7  = 

3= 

4= 

M 

3s 
?= 


M° 


ABODE 

A.  B  and  C-U.  S.  Coast  Survey.        D— IT.  S.  I.ake  Survey.        E-U.  S.  Engineers. 
Fig.  93. 


PLANE   SURVEYING 


which  the  rodmaii  can  tell  whether  the  rod  is  in  a  vertical  plane. 

Figs.  90  and  93,  D,  show  two  types  of  graduations  suitable 
where  the  meter  is  the  unit.  Fig.  93,  D,  has  for  many  years  been 
used  by  the  United  States  Coast  and  Geodetic  Survey  as  well  as 
by  the  United  States  Lake  Survey.  The  angles  of  graduation  divide 
the  rod  into  two  centimeter  intervals.  Fig.  90  shows  the  rod  used 
011  the  survey  of  the  Mexican  border.  The  graduation  is  apparent, 
and  110  further  explanation  need  here  be  given. 

Figs.  92  and  93,  C,  are  types  suitable  where  the  foot  is  the  unit. 
In  Fig.  92  the  width  comprised  between  the  ends  of  the  points  divide 
into  five  equal  parts,  the  vertical  black  lines  taking  up  two  of  these 
differences.  The  diagonal  then  gives  one  hundredth  of  a  foot,  and 
permits  readings  direct  to  single  feet.  Fig.  91  shows  a  plain 
rod  without  scale,  and  the  unit  is  the  foot.  Classes  A,  B,  C,  D  and 
E,  Fig.  93,  belong  to  the  respective  surveys  as  indicated. 


130 


PLANE   SURVEYING 


TIME  OF  ELONGATION  AND  CULMINATION  OF  POLARIS. 


BATE  IN  1899. 

EAST 
ELONGATION. 

UPPER 
CULMINATION. 

WEST 
ELONGATION. 

LOWER 
CULMINATION. 

h. 

m. 

h. 

m.            h. 

m. 

h. 

m. 

January 

1 

0 

41.9 

6 

36.7 

12 

31.5 

18 

34.7 

15 

23 

42.7 

5 

41.7 

11 

36.2 

17 

39.4 

February 

1 

22 

35.5 

4 

34.3 

10 

29.1 

16 

32.3 

15 

21 

40.3 

3 

39.0 

9 

33.9 

15 

37.0 

March 

1 

20 

45.1 

2 

43.6 

8 

38.6 

14 

41.8 

15 

19 

50.0 

1 

48.8 

7 

43.5 

13 

46.8 

April 

1 

18 

43.0 

0 

41.7 

6 

36.5 

12 

39.8 

15 

17 

48.0 

23 

42.8 

5 

41.5 

11 

44.8 

May 

1 

1C 

45.2 

22 

39.9 

4 

38.7 

10 

41.9 

15 

15 

50  3 

21 

45.0 

3 

43.8 

9 

47.0 

June 

1 

14 

43.6 

20 

38.4 

2 

37.1 

8 

40.4 

15 

13 

48.7 

19 

43.5 

1 

42.2 

7 

45.5 

July 

1 

12 

46.1 

18 

40.9 

0 

39.6 

6 

42  9 

15 

11 

51.2 

17 

46.0 

23 

40.8 

5 

48.0 

August 

1 

10 

44.7 

16 

39.5 

22 

34.3 

4 

41.5 

15 

9 

49.8 

15 

44.6 

21 

39.4 

3 

46.6 

September 

1 

8 

43.2 

14 

38.0 

20 

32.8 

2 

40.0 

15 

7 

48.3 

13 

43.1 

19 

37.9 

1 

45.1 

October 

1 

6 

45.5 

12 

40.3 

18 

35.1 

0 

42.3 

15 

5 

50.5 

11 

45.3 

17 

40.1 

23 

43.4 

November 

1 

4 

43.7 

10 

38.5 

16 

as.  3 

22 

36.5 

15 

3 

48.5 

.     9 

43.3 

15 

38.1 

21 

41.3 

December 

1              2 

45.5 

8 

40.3 

14 

.35.1 

20 

38.3 

15 

j 

50.2 

7 

45.0 

13 

39.8 

19 

43.0 

AZIMUTH  OF  POLARIS  AT  ELONGATION. 


YEAR. 

25° 

30- 

.35  r     40 

45' 

50° 

55° 

1900 

1°21'  .2  1°24'  .9 

1°  29'  .8  1°  36'  .0 

1°  44'  .0 

1°  54'  .4 

0°08'  .3 

1901 

1  20  .81  24  .6 

1  29  .4  1  35  .6 

1  43  .6 

1  54  .0 

2  07  .8 

1902 

1  20  .5  1  24  .2 

1  .29  .0  1  35  .2 

1  43  .2 

1  53  .5 

2  97  .2 

1903 

1  20  .1 

1  23  .9;  1  '28  .7 

1  34  .8 

1  42  .7 

1  53  .0 

2  06  .6 

1904 

1  19  .8,1  23  .5  1  28  .3 

1  34  .4 

1  42  .3 

1  52  .5 

2  06  .1 

1905 

1  19  .4  1  23  .1  1  27  .9 

1  34  .0 

1  41  .8 

1  52  .0 

2  05  .6 

1906 

1  18  .4 

1  22  .1 

1  26  .8 

1  32  .8 

1  40  .5 

1  50  .0 

2  05  .0 

1907 

1  18  .7 

1  22  .4 

1  27  .1 

1  .33  .2 

1  40  .9 

1  51  .0 

2  04  .4 

1908 

1  19  .4 

1  22  .1  (  1  26  .8 

1  32  .8 

1  40  .5 

1  50  .6 

2  03  .9 

PLANE  SURVEYING 


181 


UNITED  STATES  GEODETIC  SURVEY 
Base  Map  of  the  United  States 

Declinations  west  of  the  line  of  zero  declination  are  — ,  those  east  are  +. 


ll 

c3  ^*" 

a  a 


I. 


H 


SJ3 

«  a 


II 


PLANE  SURVEYING. 

PART  III. 


THE  QRADIENTER. 

The  vertical  circle  or  arc  of  the  transit  or  theodolite,  under 
ordinary  circumstances,  furnishes  the  means  of  measuring  the 
vertical  angle  through  which  the  line  of  collimation  is  turned,  or, 
on  the  other  hand,  of  turning  the  line  of  collimation  through  any 
desired  vertical  angle.  Much  of  the  work  of  the  engineer  consists 
in  measuring  slopes  or  grades,  or  in  setting  a  line  at  a  certain 
slope  or  grade;  and  the  data  are  given,  not  in  terms  of  the  vertical 
angle  directly,  but  usually  by  the  amount  of  rise  or  fall  per  100 
feet.  Thus,  a  rise  or  fall  of  2  feet  in  100  feet  is  designated  as  a 
2  per  cent  grade;  a  rise  or  fall  of  50  feet  to  the  mile  would  be 
designated  as  a  0.95  per  cent  grade,  etc.  The  ratio  of  these  two 
quantities,  rise  (ovfall}  to  reach  is  evidently  the  natural  tangent 
of  the  angle  of  slope;  and  before  the  vertical  circle  can  be  used  for 
setting  off  such  slopes,  the  ratio  must  be  transformed  into  degrees 
and  minutes  of  arc. 

The  tangent-screw  of  the  horizontal  axis  of  the  telescope, 
without  the  aid  of  the  vertical  circle,  provides  the  means  of  quick- 
ly and  accurately  setting  off  slopes  directly,  when  the  vertical 
angle  does  not  exceed  fifteen  or  twenty  degrees.  For  this  pur- 
pose, the  ordinary  tangent-screw  is  replaced  by  a  fine  screw,  with 
very  uniform  pitch  and  large  graduated  head,  and  also  a  grad- 
uated scale  from  which  may  be  read  the  number  of  turns  or  double 
turns  made  by  the  screw.  The  graduated  head  fits  friction -tight 
upon  the  neck  of  the  screw,  so  that  its  index  may  be  made  to  read 
zero  when  the  line  of  collimation  is  horizontal;  and  it  is  usually 
divided  into  fifty  parts,  so  that,  after  the  number  of  double  turns 
is  read  from  the  scale,  it  will  give  the  number  of  fiftieths  of  a 
single  turn,  or  hundredtha  of  a  double  turn.  (See  Fig.  94.) 

Let  the  distance  of  the  screw  from  the  axis  about  which  the 
telescope  turns,  be  represented  by  I  and  the  interval  between 

Copyright,  1908,  by  American  School  of  Corrtsfondence. 


PLANE    SURVEYING 


the  threads  by  t.  If  the  screw  is  turned  through  one  revolution, 
the  lever  AD  (Fig.  94)  is  moved  through  the  distance  DE,  and  the 
line  of  collimation  through  the  distance  BC,  upon  the  rod  PQ. 

T)"P  RO  / 

Xow,  the  tangent  of  the  angle  DAE  =  _— --  =  -~^-  =  — .     To 

A- 1-/          A.D  (/ 

this  ratio,  the  maker  of  the  instrument  can  give  any  convenient 
value,  but  it  is  customary  to  make  it  —  -g^,  and  it  will  be  so  con- 
sidered throughout  this  discussion. 

If,  then,  the  line  of  collimation  be  directed  toward  the  gradu- 
ated rod    PQ,  the  space  over  which  the  line  of  collimation    is 

moved  for  one  revolution  of  the 

screw  is    -(   '-^  of  the  distance  of 
(Ci'U 

the  rod  from  the  instrument; 
and  the  space  upon  the  rod  over 
which  it  is  moved  for  two  rev- 


Fig.  94. 


olutions  of  the  screw  =  -n^  = 

1 
YTTTv-  of  the  above  distance.     If 

the  screw  is  turned  through  less  than  a  single  revolution,  it  will  be 
indicated  upon  the  graduated  head,  as,  for  instance,  -p  A  of  a  single 

71  7 

turn,  the  intercept  upon  the  rod  being  ^  X-STTTV  —   9  nnA°^  ^ue 

distance  from  rod  to  instrument — it  being  understood,  of  course, 
that  the  rod  is  held  perpendicular  to  the  line  of  collimation  in  its 
initial  position.  The  index  of  the  graduated  head  should  read 
zero  when  the  line  of  collimation  is  horizontal,  and  the  reading  of 
the  scale  of  revolutions  should  be  zero  at  the  same  time. 

The  gradienter  may  be  used  as  a  telemeter,  as  a  level,  or 
simply  as  a  grade-measurer,  as  will  be  explained  in  what  follows. 

Call  s  the  intercept  upon  the  rod  for  any  movement  of  the 
gradienter-screw,  and  d  the  distance  from  the  instrument  to  the  rod. 

If  the  number  of  revolutions  of  the  gradienter-screw  is  known, 
whether  s  and  d  are  known  or  not,  the  tangent  of  the  angle  of 
inclination  of  the  line  AC  is  known,  and  the  instrument  is  a  grade- 
measurer  or  gradienter. 


PLANE    SURVEYING  135 

If  the  space  s  and  the  number  of  revolutions  of  the  gradienter- 
screw  are  known,  the  distance  d  is  known,  and  the  instrument  is 
then  a  telemeter. 

If  the  distance  d  and  the  number  of  revolutions  of  the  gradi- 
enter-screw  are  known,  the  space  s  is  known,  and  the  instrument 
then  serves  the  purpose  of  a  level. 

As  a  gradienter,  the  instrument  may  be  used  either  to  meas- 
ure the  grade  of  a  given  line,  or  to  lay  out  a  line  to  a  required 
grade. 

(1.)  Let  AB  (Fig.  05)  be  the  line  whose  grade  is  required. 
Set  the  transit  up  over  the  point  A,  and  level  carefully.  Measure 
the  height  of  the  cross-hairs  above  the  ground  by  holding  the  rod 
beside  the  instrument  and  noting  the  point  upon  the  rod  directly 
opposite  the  center  of  the  horizontal  axis  of  the  telescope.  Bring 
n  the  line  of  collimation  CE  hori- 

zontal by  means  of  the  bubble 
attached  to  the  telescope  (the 
instrument  is  supposed  to  be  in 
adjustment),  and  set  both  the 
indexes  to  zero.  Now  carry  the 

rod  to  the  point  B,  and  by  means  of  the  gradienter-screw  turn  the 
telescope  in  a  vertical  plane  until  the  line  of  collimation  strikes  the 
point  D  as  far  above  B  as  C  was  above  A.  Kow  count  the  number 
of  full  turns  from  the  reading  of  the  scale  by  the  screw-head,  and 
the  number  of  fractions  of  a  turn  from  the  divided  head.  The 
former  will  give  the  rise  (or  fall)  infect  per  100,  and  the  latter  in 
hundredths  of  a  foot.  It  must  be  remembered  that  if  the  screw  has 
made  more  than  a  whole  turn  past  the  last  number  on  the  scale 
the  reading  of  the  head  must-be  increased  by  fifty. 

Thus,  if  the  reading  of  the  scale  is  3  and  the  reading  of  the 
head  is  35,  plus  one  whole  revolution,  the  rise  (or  fall)  per  100 
feet  will  be  as  follows: 

3    double  turns    -----    3.00  ft. 

1     single  turn 0.50  ft. 

35       "  '  0.35  ft. 

6  0  - 

3.85  ft. 

So  that  the  slope  of  CD,  which  equals,  that  of  AB,  is  therefore 
3.85  per  cent. 


136 


PLANE    SURVEYING 


EXAMPLE  FOR  PRACTICE. 

If  the  scale  reads  2  and  the  head  31,  determine  the  percent- 
age of  slope. 

(2.)  It  is  required  to  layout  in  a  given  direction  a  line  with 
a  given  percentage  of  slope  from  the  point  A.  See  Fig.  96. 

Set  up  the  instrument  on  the  given  point,  as  B,  and  level  it 
carefully.  Measure  the  height  of  the  cross-hairs  above  the  ground 
as  before,  and  set  the  pointers  to  read  zero  with  the  bubble  in  the 
center  of  the  telescope  tube.  Now  revolve  the  line  of  collimation 

in  a  vertical  plane  by  means  of 
the  gradienter-screw  BO  as  to  set 
off  the  required  elope.  For  in- 
stance, suppose  it  is  required  to 
set  off  a  elope  of  2.78  per  cent. 
The  screw  should  be  turned 
five  complete  revolutions  as  in- 
dicated upon  the  scale,  plus  |* 

of  a  revolution  as  indicated  upon  the  divided  head  of  the  screw: 
5 


X  100  ft. 


=  2.50  feet. 
=  0.28  feet 


=  2.78  feet  per  100  feet. 

Now  carry  the  rod  to  any  convenient  point,  as  G,  in  the  direction 
of  the  required  line;  hold  it  in  a  vertical  position;  and  note  the 
height  of  the  line  of  collimation.  Take  the  difference  between 
this  and  CE  (.—  AB).  If  this  difference,  as  EG,  is  positive,  it 
gives  the  height  of  the  grade  line  *zlove  the  ground  and  indicates 
a  Jill  at  the  point.  If  the  difference  is  negative,  as  DE,  it  gives 
the  depth  of  the  grade  line  below  the  ground  and  indicates 


EXAMPLE  FOR  PRACTICE. 

Let  it  be  required  to  set  off  a  3.35  per  cent  grade,  and 
describe  the  operation  in  detail. 

When  the  gradienter  is  used  as  a  telemeter,  it  may  be  upon 
level  or  sloping  ground. 

(1.)  Upon  Level  Ground.  Set  up  the  transit  at  one  end  of 
the  line,  and  level  carefully.  Bring  the  bubble  of  the  telescope 
level  to  the  center  of  its  tube,  and  both  gradienter  scales  to  zero. 


PLANE    SURVEYING 


137 


Now  send  the  rod  to  the  next  station   and  let  it  be  held  vertical; 
adjust  the  target  to  the  line  of  collimation  and  take  the  reading. 
Now  turn  the  gradienter-screw  through  two  revolutions  and  take 
the  reading  again  (see  Fig.  97). 
The  difference  of  the  two  read- 
ings gives  DE  in  feet;  and  since 
the   gradienter-screw   has  been 
turned  through  two  revolutions, 
CE  =  100  DE.     Thus,  if  DE 
=  3.25  feet,  CE  =  325  feet. 

(2.)  Upon  Sloping  Ground.  On  sloping  ground  the  first 
reading  upon  the  rod  cannot  be  taken  with  the  telescope  horizon- 
tal, but  the  telescope  must  be  revolved  in  a  vertical  plane  until  the 
intersection  of  the  cross-hairs  falls  at  a  division  upon  the  rod 
equal  to  the  height  of  the  cross-hairs  above  the  ground  at  the 
transit  station.  If  now  the  rod  be  held  perpendicular  to  the  line 
of  sight,  and  the  gradienter-screw  turned  through  two  revolutions, 

the  intercept  upon  the  rod  will  be  y^-  of   the  required    distance. 

With  the  gradienter,  as  with  the  stadia,  it  is  more  convenient  to  hold 
the  rod  vertical  and  apply  the  necessary  correction  to  the  rod  reading. 
Set  up  the  transit  over  a  point  at  one  end  of  the  line  and  level 
carefully.  Measure  the  height  of  the  cross-hairs  above  the  ground. 
Now  loosen  the  clamp  of  the  tangent-screw  attached  to  the  vertical 

arc  or  circle,  and  revolve  the 
telescope  in  a  vertical  plane  until 
the  intersection  of  the  cross- 
hairs falls  upon  a  point  C  (Fig. 
98)  upon  the  rod  held  at  D,  such 
that  CD  —  AB.  Read  and  note 
the  vertical  angle  after  clamping 
the  gradienter-screw  with  both  scales  set  to  read  zero.  This  angle 
will  be  0  (Fig.  98).  Now  turn  the  gradienter-screw  through 
two  revolutions,  and  note  the  reading  ED  upon  the  rod;  the  differ- 
ence between  this'  and  CD  (=  AB)  will  give  EC,  which  call  S'; 
let  FC,  the  perpendicular  intercept  upon  the  rod,  be  called  S;  the 
distance  AC,  parallel  to  the  slope,  IT';  and  let  the  horizontal  dis- 
tance AG  be  denoted  by  II.  Then  from  the  figure, 


Pig.  98. 


138  PLANE    SURVEYING 

H'  =  100  S. 

Now,  from  the  right  triangle  EAG,  the  angle  at  E  =  90°  - 
(  $  +  <£);  and  from  the  right  triangle  FAC,  the  angle  at  F  =  90° 
-  0.  Therefore,  in  the  triangle  CFE,  the  angle  at  F  =  180°  - 

(90°  -6)  =  90°  -f  6. 

Therefore, 

S  :  S'  :  :  sin  [90°  -  (B  +  <£)]  :  sin  (90  +  0); 
or  S  :  S'  :  :  cos  (6  +  <f>]  :  cos  0. 

Hence  S  =  S-C08('  +  *) 


cos  6 

cos  0  cos  d>  -  sin  0  sin  <f> 
=  b  —    -r— -  -  =  b'  (cos  9  -  sin  (f>  tan  0): 


but  tan  0  ~  T/ITT  ?  an<^  therefore  S  =  S'  (cos  <f>  -  sin  <f>  =•  ,-^A 
1UU       •  J.UU ' 

Therefore  II',  the  distance  along  the  slope, 

=   S'  (100  cos  (j>  -  sin  </>) ; 
and  H,  the  horizontal  distance, 

=  II'  cos  <j>  =  S'  (100  cos2  (f>  -  cos  <f)  sin  0) 
=  S'  (100  cos2  cf>  -  $  sin  2  <£) 
=  100  S'  -  S'  (100  sin2  <£ -I- 1  sin  2  0). 

It  may  be  well  to  note  that  the  lower  reading  of  the  rod  need  not  nec- 
essarily be  such  as  to  make  CD  =  AB,  but  only  as  a  matter  of  convenience. 

EXAMPLE  FOR  PRACTICE. 

Upper  rod  reading  =  7.49 
Lower  rod  reading  =  4.67 
Vertical  angle  of  lower  rod  reading  =  15°  35'. 
Required  to  find  the  distance  parallel  to  the  slope  between  B  and 
D  and  the  horizontal  distance  AG. 

Let  it  now  be  required  to  find  the  difference  of  elevation 
between  B  and  D  =  CG. 

Evidently  CG  =  II  tan  (j>  =  S'  (100  cos2  <j>  tan  <£  -  cos  0  sin 
</>  tan  (£). 

=  S'  (100  sin  <f>  cos  <f>  -  sin2  <f>] 
=  S'  (100  I  sin  2  0  -  sin2  <j>\ 

In  the  last  example,  determine  the  difference  of  level  of  B 
and  D. 

It  must  not  be  forgotten  of  course,  that  the  gradienter  used 
in  this  way  cannot  give  results  eo  accurately  as  the  spirit  level; 


PLANE    SURVEYING  139 

but  nevertheless,  for  rapid  work,   the  results  will  be  sufficiently 
correct. 

If  the  student  possesses  a  set  of  stadia  reduction  tables,  the 
values  of  sin2  <f>  and  i  sin  2  <j>  can  be  taken  out  at  once  and  much 
labor  saved. 

To  Lay  out  a  Meridian  with  the  Transit.  By  means  of  the 
North,  Star  at  Upper  or  Lower  Culmination.  Twice  in  24  hours 
(more  exactly,  23  hours  56  minutes)  the  north  star  "culminates"; 
that  is  to  say,  it  attains  to  its  maximum  distance  from  the  pole,  above 
or  below  it.  At  the  moment  of  culmination,  the  star  is  upon  the 
meridian  and  if,  therefore,  a  line  be  ranged  out  upon  the  ground  in 
the  same  vertical  plane,  it  will  define  a  meridian. 

Set  up  the  transit  over  a  peg,  in  an  open  space,  giving  an 
unobstructed  view  of  a  line  about  400  or  500  feet  long.  Level  the 
instrument  carefully  (it  should  be  in  perfect  ad  justment),  and,  a  few 
minutes  before  the  time  of  culmination,  as  given  in  the  table,  focus 
the  intersection  of  the  cross-hairs  upon  the  star ;  clamp  the  plates,  the 
vertical  axis,  and  the  horizontal  axis  of  the  telescope.  Now  by 
means  of  the  tangent-screws  attached  to  the  vertical  axis  and  to  the 
vertical  circle,  move  the  telescope  in  azimuth  and  altitude,  keeping 
the  cross-hairs  fixed  upon  the  star.  After  a  time  it  will  be  found 
that  the  position  of  the  star  no  longer  changes  in  altitude;  it  is 
then  upon  the  meridian.  Now  clamp  the  vertical  axis,  plunge  the 
telescope,  and  carefully  center  a  stake  400  or  500  feet  from  the 
instrument;  the  line  connecting  the  two  stakes,  will  define  the  true 
meridian.  , 

The  whole  operation  may  be  repeated  several  nights  in  sue. 
cession,  and  the  mean  of  all  the  results  taken. 

By  Means  of  the  North  Star  at  Eastern  or  Western  Elon- 
gation. Twice  in  24  hours,  the  north  star  attains  to  its  maximum 
distance  east  or  west  of  the  pole,  called  its  eastern  or  western  "  elon- 
gation."  If  a  line  be  ranged  out  upon  the  ground  in  the  direc- 
tion of  the  star — at,  say,  the  time  of  eastern  elongation,  and  again 
at  the  time  of  western  elongation — and  if  the  angle  between 
these  two  lines  be  bisected  by  a  third  line,  this  last  line  will  evi- 
dently be  a  true  north  and  south  line. 

Otherwise.  Having  laid  out  a  line  upon  the  ground  in  the 
direction  of  the  north  star — say  at  western  elongation — take  from 


140  PLANE    SURVEYING 


a  table  the  azimuth  (or  bearing)  of  the  star  at  such  time,  and  upon 
the  horizontal  plate  of  the  transit  set  off  this  angle  to  the  east  and 
range  out  a  line — which  will  therefore  be  a  true  north  and  south 
line.  If  the  position  of  the-star  is  taken  at  eastern  elongation,  the 
azimuth  must  be  turned  off  to  the  west. 

Set  up  the  transit  over  a  peg  a  few  minutes  before  the  star 
attains  its  maximum  elongation,  as  given  by  the  table.  Level,  and 
fix  the  line  of  collimation  upon  the  star,  following  its  movement 
as  described  under  the  previous  method.  After  a  time,  it  will  be 
found  that  the  movement  of  the  star  in  azimuth  ceases;  the  star 
has  then  attained  its  maximum  elongation.  Now  clamp  the  ver- 
tical axis  of  the  instrument,  plunge  the  telescope,  and  center  a  stake 
in  the  proper  direction.  Now  take  from  the  table  the  proper  azi- 
muth, revolve  the  upper  plate  through  the  given  angle  in  the 
proper  direction,  and  range  out  a  line  upon  the  ground  for  the  true 
meridian. 

In  order  to  determine  the  azimuth  of  the  north  star  at  eastern 
or  western  elongation,  it  is  necessary  to  know  the  latitude  of  the 
place  of  observation. 

Definitions.  The  altitude  of  a  star  is  the  vertical  angle  at 
the  instrument  included  between  the  plane  of  the  horizon  and  the 
line  from  the  instrument  to  the  star,  as  given  by  the  line  of  colli- 
mation. 

The  latitude  of  a  place  is  equal  to  the  altitude  of  the  pole. 

If,  therefore,  we  have  any  method  of  determining  the  altitude 
of  the  pole,  the  latitude  of  the  observer  is  known  at  once. 

The  altitude  of  the  pole  may  be  determined  by  observing  the 
altitude  of  the  north  star,  first  at  its  upper  culmination  and  again 
at  its  lower  culmination.  The  mean  of  these  observations,  cor- 
rected for  refraction,  will  give  the  altitude  of  the  pole  and  there- 
fore the  latitude  of  the  observer.  See  tables  of  refraction  of  Polaris. 

Set  up  the  transit  and  level  it,  and  proceed  in  the  eame  man- 
ner as  described  under  the  first  method  for  laying  out  a  true  merid- 
ian. When  the  star  has  reached  its  maximum  distance  above  or 
below  the  pole,  as  indicated  by  the  line  of  collimation  moving  in 
a  horizontal  plane,  clamp  the  horizontal  axis  of  the  telescope  and 
read  the  angle  upon  the  vertical  circle.  The  result  will  be  the 
altitude  of  the  star,  say  at  upper  culmination.  Repeat  the  operation 


PLANE    SURVEYING 


141 


at  lower  culmination.  Now,  if  A  represents  the  altitude  at  upper 
culmination,  and  Al  the  altitude  at  lower  culmination ;  d  the  refrac- 
tion at  upper  culmination,  and  dl  the  refraction  at  lower  culmina- 
tion, then  Ap,  the  altitude  of  the.  pole  (=  latitude  of  the  place), 
will  be  given  by  the  following: 

.  Ap  =  i(A  +  Ai-d-4) 

It  will  be  well  to  repeat  these  observations  and  take  the  mean 
of  the  results  as  the  probable  altitude  of  the  pole. 


Fig.  99. 


THE  SOLAR  TRANSIT. 

The  solar  transit  is  an  ordinary  engineer's  transit  fitted  with 
a  solar  attachment.  Of  the  many  forms  of  solars  in  use,  that 
invented  by  G.  N.  Saegmuller,  Washington,  D.  C.,  seems  to  be  the 
favorite.  In  its  latest  form  it  is  shown  in  Fig.  99,  and  consists  of 
a  telescope  and  level  attached  to  the  telescope  of  the  transit  (see 
Fig.  100)  in  such  a  manner  as  to  be  free  to  revolve  in  two  direc- 
tions at  right  angles  to  each  other.  When  the  transit  telescope  is 
horizontal  and  the  bubble  of  the  solar  in  the  center  of  its  tube,  the 
tmxiliary  telescope  with  its  bubble  revolves  in  horizontal  and  ver- 
tical planes. 


142 


PLANE    SURVEYING 


PLANE    SURVEYING  143 

If  now  the  line  of  collimation  of  the  transit  be  brought  into 
the  meridian,  the  telescope  pointing  to  the  south,  then,  if  we  lay 
off  upon  the  vertical  circle,  upward,  the  co-latitude  of  the  place, 
the  polar  axis  of  the  solar  will  be  parallel  to  the  axis  of  the  earth. 
If  now  the  two  lines  of  sight  are  parallel  and  the  solar  telescope  is 
revolved  upon  its  polar  axis,  it  is  evident  that  its  line  of  sight  will 
describe  a  plane  parallel  to  the  plane  of  the  equator.  If  now  the 
transit  telescope  be  still  maintained  parallel  to  the  equator,  if  we 
turn  the  Bolar  telescope  upon  its  horizontal  axis  until  the  angle 
between  the  two  lines  of  collimation  equals  the  declination  of  the  sun, 
then  when  the  solar  telescope  is  revolved  upon  its  polar  axis,  its 
line  of  collimation  will  follow  the  path  of  the  sun  for  the  grven 
day,  provided  there  be  no  change  in  the  sun's  declination.  If 
therefore  the  solar  telescope  is  revolved  until  the  image  of  the  sun 
is  brought  between  a  pair  of  horizontal  and  vertical  wires,  pro- 
vided in  the  telescope  for  that  purpose,  at  that  instant  the  line  of 
sight  of  the  transit  telescope  is  in  the  meridian. 

The  horizontal  axis  of  the  solar  telescope  and  the  polar  axis  of 
the  Bolar  are  provided  with  clamps  and  tangent-screws  by  means 
of  which  careful  adjustments  may  be  made..  Two  pointers  are  at- 
tached to  the  solar  telescope,  so  adjusted  that  when  the  shadow  of 
the  one  is  thrown  upon  the  other,  the  sun  will  appear  in  the  field 
of  view.  There  are  also  provided  colored  glass  shade's  to  the  eye- 
piece to  protect  the  eye  when  observing  upon  the  snn.  The 
objective  and  the  cross-hairs  are  focused  in  the  usual  way. 

Adjustments  of  the  Solar  Transit.  It  is  assumed  in  what 
follows  that  the  transit  is  in  perfect  adjustment,  particularly  the 
plate  levels,  the  horizontal  axis  of  the  telescope,  and  the  zero  of 
the  vertical  circle. 

1.  To  adjust  the  Polar  Axis.  The  polar  axis  should  be 
vertical  when-  the  line  of  collimation  and  the  horizontal  axis  of  the 
telescope  are  horizontal.  To  make  this  adjustment,  level  the  tran- 
sit by  means  of  the  plate  levels.  If  the  telescope  is  not  fitted  with 
a  level,  make  the  vernier  of  the  vertical  circle  read  zero.  Now 
bring  the  bubble  of  the  solar  to  the  center  of  its  tube  and  clamp 
the  horizontal  axis.  Loosen  the  clamp  of  the  polar  axis,  and  turn 
the  solar  upon  its  polar  axis  through  180°.  If  the  bubble  remains 
in  the  center  of  the  tube,  the  solar  axis  is  in  adjustment.  If  the 


144  PLANE    SURVEYING 


bubble  runs  toward  one  end  of  the  tube,  correct  one-half  of  the 
error  by  revolving  the  solar  telescope  upon  its  horizontal  axis  and 
the  other  half  by  means  of  the  capstan -headed  screws  at  the  base 
of  the  solar. 

If  the  telescope  of  the  transit  is  fitted  with  a  level,  it  will  be 
better  to  test  the  vertically  of  the  vertical  axis  by  means  of  it, 
since  it  is  longer  and  more  sensitive  than  the  bubbles  upon  the 
plate.  To  do  this,  revolve  the  telescope  upon  ita  vertical  axis 
until  it  is  directly  over  a  pair  of  diagonally  opposite  plate  screws, 
and  bring  the  bubble  to  the  center  by  means  of  the  tangent-screw 
attached  to  the  horizontal  axis  of  the  telescope.  Now  revolve  the 
telescope  upon  its  vertical  axis  through  180°,  and  note  if  the  bub- 
ble runs  to  one  end;  if  it  does  correct  one-half  the  error  by  the 
parallel  plate-screw  and  the  other  half  by  the  tangent-screw  of  the 
horizontal  axis,  and  repeat  this  test  and  correction  until  the  bubble 
remains  in  the  center  in  all  positions. 

2.  To  Adjust  the  Cross -Hairs  of  the  Solar.  The  line  of 
collimation  of  the  solar  telescope  should  be  parallel  to  the  line  of 
collimation  of  the  transit  telescope.  The  first  adjustment  having 
been  made,  first  bring  the  telescope  into  the  same  vertical  plane  by 
centering  a  stake  by  the  transit  telescope  and  clamping  the  verti- 
cal axis.  Now  turn  the  telescope  of  the  solar  upon  the  polar  axis 
until  the  intersection  of  the  cross-hairs  covers  the  same  point  upon 
the  stake,  and  clamp  the  polar  axis.  Now  level  both  telescopes 
by  bringing  the  bubbles  to  the  center,  and  measure  the  distance 
between  the  axes  of  the  two  telescopes;  draw  at  this  distance  two 
black  parallel  lines  upon  a  piece  of  white  paper.  Tack  up  the 
paper  against  a  wall,  post,  or  other  convenient  object,  adjusting  it 
in  position  so  that  one  black  line  is  covered  by  the  horizontal  cross- 
hair  of  the  transit  telescope;  notice  if  the  other  black  line  is  cov- 
ered by  the  horizontal  cross-hair  of  the  solar;  if  so,  the  adjustment 
is  completed;  otherwise,  move  the  diaphragm  carrying  the  cross- 
hairs of  the  solar,  until  the  second  black  line  is  covered.  Adjust- 
ing the  cross-hair  diaphragm  may  displace  the  solar  telescope  ver- 
tically, so  that  the  bubble  should  again  be  brought  to  the  center  of 
the  tube,  and  the  adjustment  tested  and  repeated  until  the  two 
lines  of  collimation  are  parallel,  when  the  two  bubbles  are  simul- 
taneously in  the  center  of  the  tubes. 


PLANE    SURVEYING  145 


The  Use  of  the  Solar  Transit.  An  observation  with  the 
solar  transit  involves  four  quantities  as  follows: 

1.  The  time  of  day,  that  is  to  say,  the  hour-angle  of  the  sun. 

2.  The  declination  of  the  sun. 

3.  The  latitude  of  the  place  of  observation. 

4.  The  direction  of  the  meridian. 

Any  three  of  these  quantities  being  known,  the  fourth  may  be 
determined  by  direct  observation.  The  principal  use  of  the  solar 
transit  is  to  determine  a  true  meridian  when  the  other  three  quan- 
titles  are  known. 

To  Lay  Out  a  True  Meridian.  Set  up  the  transit  over  a  stake; 
level  the  instrument  carefully;  and  bring  the  lines  of  collimation 
of  the  telescopes,  into  the  same  vertical  plane  by  the  method  pre- 
viously described.  Take  the  declination  of  the  sun  as  given  in  the 
Nautical  Almanac  for  the  given  day,  and  correct  it  for  refraction 
and  hourly  change.  Revolve  the  transit  telescope  upon  its  hori- 
zontal axis  so  that  the  vertical  circle  will  record  this  corrected  dec- 
lination, turning  it  down  if  the  declination  ia  north,  and  elevating 
it  if  the  declination  ia  south.  Now,  without  disturbing  the  posi- 
tion of  the  transit  telescope,  bring  the  solar  telescope  to  a  horizon- 
tal position  by  means  of  the  attached  level.  ^  It  is  evident  that  the 
angles  between  the  lines  of  collimation  will  equal  the  corrected 
declination  of  the  Bun,  and  the  inclination  of  the  solar  telescope 
to  its  polar  axis  will  be  equal  to  the  polar  distance  of  the  sun. 

Next,  without  disturbing  the  relative  positions  of  the  two  tele- 
scopes,  set  the  vernier  of  the  transit  telescope  to  the  co-latitude  of 
the  place,  and  clamp  the  horizontal  axis.  It  is  evident  that  the 
transit  telescope  is  parallel  to  the  equator,  and  that  the  solar  tele- 
scope is  in  a  position  to  describe  the  path  of  the  sun  when  the  line 
of  collimation  of  the  transit  is  in  the  true  meridian;  and  unless 
the  line  of  collimation  is  in  the  true  meridian,  the  sun  cannot  be 
brought  between  the  cross-hairs  of  the  solar  telescope.  Therefore 
unclamp  the  vertical  axis  of  the  transit  and  the  polar  axis  of  the 
solar,  and,  maintaining  the  relative  positions  of  the  telescopes 
revolve  the  transit  upon  its  vertical  axis,  and  the  solar  upon  its 
polar  axis,  until  the  sun  is  brought  between  the  cross-hairs  of  the 
solar  telescope.  Now  clamp  the  vertical  axis  of  the  transit  and 
range  out  a  line  upon  the  ground  for  the  true  meridian. 


146  PLANE    SURVEYING 

The  solar  apparatus  should  not  be  used  between  11  A.  M.  and 
1  p.  M.  if  the  best  results  are  desired.  From  7  to  10  A.  M.  and 
from  2  to  5  P.  M.  in  the  summer  will  give  the  best  results.  The 
greater  the  hour-angle  of  the  sun,  the  better  the  observation  will 
be  so  far  as  instrumental  errors  are  concerned.  However,  if  the 
sun  is  too  close  to  the  horizon,  the  uncertainties  in  regard  to  refrac- 
tion will  cause  unknown  errors  of  considerable  magnitude. 

Observation  for  Time.  If  the  two  telescopes — being  in 
position,  one  in  the  meridian  and  the  other  pointing  to  the  eun — 
are  now  revolved  upon  their  horizontal  axes  (the  vertical  remaining 
undisturbed)  until  each  is  level,  the  angle  upon  the  horizontal 
plate  between  their  directions,  as  found  by  sighting  on  a  distant 
object,  will  give  the  time  from  apparent  noon,  reliable  to  within  a 
few  seconds. 

To  Determine  the  Latitude.  Level  the  transit  carefully,  and 
point  the  telescope  toward  the  south,  setting  off  the  declination  of 
the  sun  upon  the  vertical  circle,  elevating  the  object  end  if  the  dec- 
lination ia  south,  and  depressing  it  if  the  declination  is  north. 
Bring  the  telescope  of  the  solar  into  the  same  vertical  plane  with 
the  transit  telescope  by  the  method  previously  described,  level  it 
carefully,  and  clamp  it.  The  angle  between  the  lines  of  collirnation 
will  then  equal  the  declination  of  the  sun.  With  the  solar  tele- 
scope, observe  the  sun  a  few  minutes  before  its  culmination,  by 
moving  the  transit  telescope  in  altitude  and  azimuth  until  the 
image  of  the  sun  is  brought  between  the  cross-hairs  of  the  solar, 
keeping  it  there  by  means  of  the  tangent- screws  until  the  sun  ceases 
to  rise.  Then  take  the  reading  of  the  vertical  circle,  correct  for 
refraction  due  to  altitude  by  the  table  below,  subtract  the  result 
from  90°,  and  the  remainder  is  the  latitude  sought. 


154 


TOPOGRAPHERS  OF  U.  S.  GEOLOGICAL  SURVEY  AT  WORK  ON  MOUNTAIN  SUMMIT 
IN  ALASKA 


PLANE    SURVEYING 


147 


Mean  Refraction  at  Various  Altitudes.* 

Barometer,  30  inches.    Fahrenheit  Thermometer,  50°. 


Altitude. 

Refraction. 

Altitude. 

Refraction. 

10° 

5' 

19" 

20' 

2' 

39" 

11 

4 

51 

25 

2 

04 

12 

4 

27 

30 

1 

41 

13 

4 

07 

35 

1 

23 

14 

3 

49 

40 

1 

09 

15 

3 

34 

45 

58 

16 

3 

20 

50 

49 

17 

3 

08                            60 

34 

18 

2 

57                            70 

21 

19                            2 

48                              80 

10 

Preparation  of  the  Declination  Settings   for  a  Day's  Work. 

The  solar  ephemeris  gives  the  declination  of  the  sun  for  the  given 
day,  for  Greenwich  mean  noon.  Since  all  points  in  America  are 
west  of  Greenwich,  by  4,  5,  G,  7,  or  8  hours,  the  declination  found 
in  the  ephemeris  is  the  declination  at  the  given  place  at  8,  7,  6,  5, 
or  4  o'clock  A.  M.  of  the  same  date,  according  as  the  place  lies  in 
the  "Colonial,"  "  Eastern."  ••Central,''  ••Mountain,"  or  "Pa- 
cific" time  belts  respectively. 

The  columns  headed  "Refraction  Corrections"  (see  table) 
give  the  correction  to  be  made  to  the  declination,  for  refraction 
for  any  point  whose  latitude  is  40°.  If  the  latitude  is  more  or  less 
than  40°,  these  corrections  are  to  be  multiplied  by  the  correspond- 
ing coefficient  given  in  the  table  of  "  Latitude  Coefficients  "  (page 
148).  Thus  the  refraction  corrections  in  latitude  30°  are  G5  one- 
hundredths,  and  those  of  50°  142  one-hundredths  of  the  correspond- 
ing ones  in  latitude  40°.  There  is  a  slight  error  in  the  use  of 
these  latitude  coefficients,  but  the  maximum  error  will  not  amount 
to  over  15  seconds,  except  when  the  sun  is  very  near  the  horizon, 
and  then  any  refraction  becomes  very  uncertain.  All  refrac- 
tion tables  are  made  out  for  the  mean  (or  average)  refraction 
whereas  the  actual  refraction  at  any  particular  time  and  place  may 
be  not  more  than  one-half  or  as  much  as  twice  the  mean  refraction, 
with  small  altitudes.  The  errors  made  in  the  use  of  these  latitude 
coefficients  are  therefore  very  small  compared  with  the  errors  re- 


*  This  table,  as  well  as  those  following,  is  taken  from  the  catalogue  of 
George  N.  Saegmuller,  Washington,  D.  C. 


148 


PLANE   SURVEYING 


suiting  from  the  use  of  the  mean,  rather  than   unknown  actual, 
refraction  which  affects  any  given  observation. 

Latitude  Coefficients. 


LAT. 

COEFF. 

LAT. 

COEFF. 

LAT. 

COEFF. 

LAT. 

OOEFF. 

15° 

.30 

27° 

.5fi 

39'-* 

.96 

51° 

.47 

16 

.32 

28 

.59 

40 

1.00 

52 

.53 

17 

.34 

29 

.62 

41 

1.04 

53 

.58 

18 

.36 

30 

.65 

42 

1.08 

54 

.64 

19 

.38 

31 

.68 

43 

1.12 

55 

.70 

20 

.40 

32 

.71 

44 

1.16 

50 

.76 

21 

.42 

33 

.75 

45 

1.20 

57 

.82 

22 

.44 

34 

.78 

46 

1.24 

58 

.88 

23 

.46 

ar> 

.82 

47 

1.29 

59 

1.94 

24 

.48 

3C 

.85 

48 

1.33 

CO 

2.00 

25 

.50 

37 

.89 

49 

1.38 

26 

.53 

38 

.92 

50 

1.42 

If  the  date  of  observation  be  between  June  20  and  September 
20,  the  declination  is  positive  and  the  hourly  change  negative; 
while  if  it  be  between  December  20  and  March  20,  the  declination 
is  negative  and  the  hourly  change  positive.  The  refraction  cor- 
rection is  always  positive;  that  is,  it  always  increases  numerically 
the  north  declination,  and  diminishes  numerically  the  south  dec- 
lination. The  hourly  refraction  corrections  given  in  the  ephem- 
eris  are  exact  each  for  the  middle  day  of  the  five-day  period,  cor- 
responding  to  that  of  hourly  corrections.  For  the  extreme  days 
of  any  such  period,  an  interpolation  can  be  made  between  the 
adjacent  hourly  corrections,  if  desired. 

By  using  standard  time  instead  of  local  time,  a  slight  error 
is  made,  but  the  maximum  value  of  this  error  is  found  at  those 
points  when  the  standard  time  differs  from  the  local  time  by  one- 
half  hour,  and  in  the  spring  and  fall  when  the  declination  is  chang- 
ing rapidly.  The  greatest  error  then,  is  less  than  30  seconds,  and 
this  is  smaller  than  can  be  set  off  on  the  vertical  circle  or  declina- 
tion arc.  Even  this  error  can  be  avoided  by  using  the  true  dif- 
ference of  time  from  Greenwich  in  place  of  standard  meridian 
time. 

EXAMPLES  FOR  PRACTICE. 

(1)  Let  it  be  required  to  prepare  a  *able  of  declination  for 
June  10,  1904,  for  a  point  whose  latitude  is  40°  20' ,  and  which 
lies  in  the  "  Central  Time"  belt. 


PLA^E    SURVEYING 


149 


Since  the  time  is  6  hours  earlier  than  that  at  Greenwich,  the 
declination  given  in  the  ephemeris  is  the  declination  at  the  given 
place  at  6  A.  M.  of  the  same  date.  This  is  found  to  be  23°  0'  18". 
To  this  must  be  added  the  hourly  chancre  which  is  also  plus  and 
equal  to  11.67".  The  latitude  coefficient  is  1.013.  The  following 
table  may  now  be  made  out. 

I  i i  j  j        i 

HOUR        DECLINATION    IREF.COR.       SF.TTINfl  HOUR    ,    DKCI.IN ATIOX     REP.  COR.          8KTTIXO 


7 

+  23°  0'  30" 

+  V  10" 

23°  1'  40" 

I 

23°  1/41" 

18" 

23°  1'  59" 

8 

4-  23°  0'  41" 

+  44" 

23°  1'  25" 

2 

23°  1'  52" 

22" 

23°  2'  14" 

9 

+  23°  0'  53" 

+  29" 

23°  1'  22", 

3 

23°  2'  04" 

29" 

23°  2'  3.T' 

10 

-j-23°r  5" 

+  22" 

22°  1'  27" 

4 

23°  2'  16" 

44" 

23°  3'  (V 

11 

+  23°  1'  17*i+  18" 

23°  lf  35" 

5 

23°  2'  28" 

1'  10" 

23°  3'  38" 

PROBLEMS  INVOLVING  USE  OF  TRANSIT. 

Perpendiculars  and  Parallels.  To  erect  ft  perpendicular  to 
a  line  at  a  given  point  of  the  line.  Set  up  the  transit  over  the 
given  point,  and  with  the  verniers  set  to  0°,  direct  the  line  of 
sight  along  the  given  line.  Clamp  the  lower  motion,  unclamp  the 
upper  motion,  and  turn  off  an  angle  of  90°  in  the  proper  direction 
for  the  required  line. 

To  erect  a  perpendicular  to  an,  inaccessible  line  at  a  given 
point  of  the  line.  Let  AB,  Fig.  101,  be  the  given  inaccessible 
line,  and  A  the  point  of  the  line  at  which  it  is  proposed  to  erect 
the  perpendicular  AD.  Select 
some  point  II  from  which  can  be 
distinctly  seen  the  points  A  and 
B  of  the  inaccessible  line.  Setup 
the  transit  at  the  point  II,  and 
measure  the  angle  AHB.  Also 
from  the  point  H  run  out  and 
measure  a  line  of  any  convenient 
length,  and  in  such  a  direction  that  the  points  A  and  B  can  be 
seen  from  its  extremity,  as  E.  Now  measure  the  angles  AHE 
and  BHE.  Now  set  up  the  transit  at  E,  and  measure  the  angles 
BEIT,  BEA,  and  AEH.  In  the  triangle  AHE,  we  know 
from  measurement  the  length  of  the  side  HE,  as  also  the  angles 
AHE  and  AEII,  from  which  may  be  calculated  the  length  of 
the  side  AH,  which  is  also  one  side  of  the  triangle  AIIB.  From 
the  triangle  BEH,  we  have  the  length  HE,  known  by  measure 


Fig.  101. 


150 


PLANE    SURVEYING 


Refraction  Correction. 

Latitude,  40°. 


January. 

February. 

March. 

April. 

May. 

Juno. 

1 

Ih.  1  58 

1 

1 

Ih.  l  o:5    :    1 

3h.  0  57 

1 

Ih.  0  28 

1 

5h.  1  11 

2      2  16 

2      1  10     !     2 

4      1  19 

2      0  32 

i 

'1 

3      3  04 

3      1  27     ;     3 

5      2  18 

2 

3      0  39 

g 

1      0  19 

8 

4      623 
1      1  54 

3 

Ih.  1  26 

1 

4      2  06 
5      4  39 

4 
5 

1      039 

2      0  44 

3 

4      0  55 
5      1  30 

1 
.7 

2      0  23 
3      0  30 

5 

1 

2      1  37 

K 

1      0  59 

6 

3      054 

i 

1      0  26 

8 

4      043 

2      2  11 

5 

3      2  04 

6 

2      1  06 

7 

4      1  14 

:, 

2      0  30 

7 

5      1  10 

7 

s 

9 
11 
18 

3      2  59 
4      601 
1      1  51 
2      207 
3      2  51 
4      540 

f> 

7 
8 
9 

JO 

11 

1J 

4      321 
1      1  21 
2      1  31 
3      1  56 
4      3  04 

7 
g 

'.i 

10 

11 

U 

13 

3      1  21 
4      1  56 
5      4  04 

1      0  55 
2      1  02 
3      1  15 
4      1  47 

8 
9 

10 

II 

U 

13 

M 

5      2  08 

1      036 
2      0  41 
3      0  51 
4      1  10 
5      1  58 
1      0  34 

7 

s 
'.« 

11 
13 

3      0  37 
4      0  53 
5      1  26 
1      0  25 
2      0  29 
3      0  36 
4      0  51 
5      1  22 

s 
li 

10 

11 
12 
13 

it 

1      018 
2      022 
3      0  29 
4      0  43 
5      1  09 
1      018 
2      0  22 

14 

1      1  46 

IS 

1      1  16 

1   1 

o       o  o4 

15 

2      0  38 

11 

1      0  23 

15 

3      029 

15 
16 

2      2  01 

ll 
IS 

3      1  48 

IS 

1C, 

0  58 

10 

n 

3      0  48 
4      1  06 

15 
16 

2      0  27 
3      0  34 

17 

4      0  42 
5      1  08 

17 

18 

s 

21 
24 

3      2  40 
4      500 
1      1  42 
2      156 
3      2  31 
4      4  35 
1      1  37 

16 

17 

IS 

lit 

JO 

Jl 

JJ 
23 

4      2  47 
5      8  39 
1      1  12 
1      1  20 
3      1  40 
4      2  31 
5      6  49 
1      1  07 

11 
19 

JO 

22 
23 
24 

1  10 

1  39 
3  08 

0  48 
0  54 
J       1  05 

19 
20 

Jl 
JJ 
23 

5      1  49 
1      032 
2      036 
3      0  45 
4      1  02 
5      142 
1      030 
2      0  34 

17 
11' 

JO 

Jl 
JJ 

83 

Jl 
85 

4      0  49 
5      1  18 
1      0  22 
2      0  26 
3      0  33 
4      0  47 
5      1  15 
1      021 
2      025 

r.* 

JO 

Jl 

JJ 

83 

ji 
25 

1      018 
2      022 
3      028 
4      0  42 
5      1  08 
1      018 
2      022 
3      029 

25 

28 

J'.i 

2      158 
3      2  22 
4      4  07 
1      1  32 

Jt 
j:. 
26 
27 

JS 

2      1  15 
3      1  33 
4      2  18 

5h.  5  28 

28 
29 

0  45 

0  50 

:     1  01 

!        2  34 

26 

28 

.„, 

so 

3      0  42 
4      0  58 
5      1  36 
1      028 
2h.  0  32 

JO 

27 

88 
89 

3      032 
4      0  46 
5      1  13 
1      020 
2      Off 

88 

L'7 

28 

j'." 

4      0  42 
5      1  08 
1      018 

2      022 
3      0  29 

2      1  44 

30 

1      0  42 

80 

4      0  31 

80 

4h.  0  43 

.')" 

3      2  13 

31 

2h.  0  47 

4      0  44 

Ml 

4h.  3  41 

31 

5h.  1  11 

July. 

August. 

September. 

October. 

November. 

December. 

j 

5h.  1  09 

1 

1 

Ih.  0  39 

1 

Ih.  0  59 

1 

2h.  3  21 

1 

Ih.  1  54 

2 

2      0  44 

2 

2      1  06 

2 

3    13  57 

2 

2      2  11 

3 
4 

5 

1      0  19 
2      0  23 
3      0  30 

3 

4 

Ih.  0  26 
2      0  30 
3      0  37 

3 
4 

3      0  54 
4      1  14 
5     208 

a 

4 

3      1  21 
4      1  56 
5      4  04 

3 
4 

4 

1      1  32 

3 
4 

3      2  59 
4      601 
5 

6 

4      0  43 

5 

4      0  53 

6 

1     0  42 

fl 

1      1  03 

,, 

2      1  44 

5 

1      1  58 

7 

5      1  10 

6 

5      1  26 

2      0  47 

| 

2         10 

7 

3      2  13 

6 

2     2  16 

8 

3      0  57 

8 

3         27 

8 

4      3  41 

^ 

3      3  04 

8 
9 

10 

1      0  20 
2      0  24 
3      0  31 

8 
9 

1      0  28 
2      0  32 
3      0  39 

9 

10 

4      1  19 
5      2  18 

'.• 

4         06 
5         39 

9 

10 

5 
1      1  37 

8 
g 

4      623 

11 

4      0  44 

10 

4      0  55 

ll 

1      0  45 

ll 

1         07 

11 

2      1  50 

10 

1      200 

12 

5      1  11 

M 

5     1  30 

2      0  50 

U 

2         15 

U 

ll 

13 

It 
15 

1      021 
2      0  25 
3      0  32 

18 

it 

1      0  30 
2      0  34 
3      0  42 

13 

ll 
15 

3      1  01 
4      1  K 
5      2  34 

13 

i: 
r, 

3         33 
4         18 
5         39 

18 

ll 

4      4  07 
5 
1      1  42 

18 

it 

3      309 
4      638 

16 

4      0  46 

4      0  58 

16 

1      048 

16 

1         12 

16 

2      1  56 

15 

1     201 

17 

5      1  13 

16 

5     1  36 

17 

2      0  54 

17 

2         20 

17 

3      2  31 

10 

2      220 

18 
19 

20 

1      022 
2      0  26 
3      0X3 

17 

is 
in 

1     032 
2      036 
3      0  45 

is 
19 

JO 

3      1  05 
4      1  32 
5      2  51 

18 
19 

JO 

3         40 
4         31 
5         29 

18 
19 

JO 

4      4  35 
5 

1      1  46 

17 

11' 

3      3  11 
4      647 
5 

21 

4      0  47 

JO 

4      1  02 

Jl 

1      0  52 

Jl 

1         16 

"1 

2      201 

JO 

1     201 

JJ 

5      1  15 

Jl 

5      1  42 

JJ 

2      0  58 

2          25 

JJ 

3      2  40 

Jl 

2      220 

23 

Jl 
25 

1      023 
2      027 
3     0  34 

..., 

23 

Jt 

1      0  34 

2      038 
3      0  48 

88 
Jl 
25 

3      1  10 
4      1  39 
5      3  08 

j.1; 
jt 
89 

3          48 
4         47 
5         39 

Jl 
25 

4      4  59 
5 
1      1  50 

88 

Jl 

3      3  11 
4      6  49 

26 

4      0  49 

4      1  06 

26 

1      055 

Jo 

1         21 

88 

2      2  06 

25 

1      200 

27 

5     1  18 

y, 

5      1  49 

2      1  02 

87 

2         31 

3      2  49 

88 

2      2  19 

88 
29 
SO 
81 

1     0  25 

2      029 
3      036 
4      051 
5h.  1  22 

•21 

JS 

j'.i 

.",0 

31 

1      036 
2      0  41 
3      0  51 
4      1  10 
5h.  1  58 

89 
80 

3      1  15 
4      1  47 
5h.  3  34 

JS 

J'.I 

80 
81 

3          56 
4      3  04 
5    11  01 
Ih.  1  26 
1  37 
2  04 

28 

J'.i 

n 

4      5  33 
5h. 

27 

JS 

J'.l 

:;i 

3      309 
4      6  43 
5h. 

PLANE    SURVEYING  151 

ment,  as  well  as  the  angles  BHE  and  BEH,  from  which  we 
can  calculate  the  length  of  the  Bide  BH,  which  is  also  one  side  of 
the  triangle  AIIB.  Therefore  in  the  triangle  AHB,  we  have 
the  lengths  of  the  two  sides  All  and  BII  by  calculation;  and  the 
angle  AIIB  by  measurement.  We  can  therefore  calculate 
the  angle  IIAB,  which  equals  the  angle  AHD.  Set  up  the  transit 
at  II,  sight  to  A,  and  turn  off  the  angle  AHD  (—  IIAB), 
measuring  off  II  D  of  a  length  equal  to  AH  cos  AHD.  Then 
AD  will  be  the  perpendicular  required,  and  its  length  will  equal 
AH  sin  AIID. 

The  calculation  is  as  follows  :     In  the  triangle  AHE,  the  angle 
HAE  =  180°  -  (AHE  +  AEII),  and  therefore  All  :  HE  :  :  sin 

AEII  :  sin  HAE;  or,  AH  =  HE  si"  f,™ 

sin  HAE 

In  the  triangle  1IEB,  IIBE  =  180°  -  (BHE  +  BEH),  and 
therefore  HB  :  HE  :  :  sin  HEB  :  sin  IIBE; 


sin  HBE 

In  the  triangle  AHB,  the  sum  of  the  angles  HAB  and  HBA 
=  180°  —  AHB.  Let  x  represent  the  difference  of  the  angles  IIAB 
and  HBA.  Then,  from  trigonometry, 

AH  +  HB  :  All  -  HB  :  :  tan  -J  (IIAB  +  HBA)  :  tan  A  (IIAB  - 
HBA); 
or,  All  +  HB  :  All      HB  ::  tan  £  (ISO1  -  AHB)  :  tan  i  .r; 


or,  AH  +  HB  :  AH-  IIB  ::  cot         -:  tan  J  ».- 

From  this  last  proportion  we  find  #,  the  difference  of  the  two  angles 
HAB  and  IIB  A.     We  then  have  the  simultaneous  equations: 

IIAB  +  IIB  A  =  c  (say) 

HAB  -   HBA  =  (I  (say) 

Therefore  IIAB  =  -^t~;  and  HBA  =  ~~. 

2  & 

EXAMPLE   FOR   PRACTICE 

Given  HE  (Fig.  101)  =  125  feet;  AHE  =  122°;  AHB  =  94°; 

BHE  =28°;  BEH  =  121°;  BE  A  =  80°;  AEII  =41°.     It  is  required 

to  find  the  angle  AHD,  the  length  of  HI),  and  the  length  of  AD 

Aus.  AHD'=  50"  54';  II  D  =  170.93  feet;  AD  =  217.92  feet 


152 


PLANE    SURVEYING 


To  let  full  a  perpendicular  to  a  line  from  a  given  point. 
Let  AB,  Fig,  102,  be  the  given  line,  and  C  the  point.  Set  up  the 
transit  at  some  point  A  of  the  given  line,  and  measure  the  angle 
BAG.  Take  the  instrument  to  C,  sight  to  A  and  turn  off  an  angle 
AOB  =  905  -  BAG.  The  instrument  will  then  sight  in  the  direc- 
tion of  the  required  perpendicular  CB. 

To  let  fall  a  perpendicular  to  a  line  from,  an,  inaccessible 
point.  LetBC,  Fig.  103,  be  the  given  line  and  A  the  inaccessible 
point  from  which  it  is  desired  to  let  fall  the  perpendicular  upon 
BC.  Set  up  the  instrument,  as  at  B;  and,  after  measuring  the 


AT- 


D 

Fig.  102.  Fig.  103. 

length  of  BC,  measure  the  angle  ABC.     Take  the  instrument  to 
G,  and  measure  the  angle  AGB.     Then  in  the  triangle  ABC, 
AB  :  BG  :  :  sin  AGB  :  sin  (AGB  +  ABC); 


and, 


BD  =  ABcos  ABC; 

tan  AGB 


BD  =  BC 


tan  ACB  +  tan  ABC' 


EXAHPLE  FOR  PRACTICE. 

Given  BC  (Fig.  103)  =  250  feet;  ABC  =  63°15';  ACB  = 
55^40'.  Calculate  the  length  of  BD,  and  the  length  of  AD. 

j  BD  =  126.2  feet. 
S'  \  AD  =  250.7  feet. 

To  let  fall  a  perpendicular  to  an  inaccessible  line  from  a 
given  point  outside  of  the  line.  Let  AB,  Fig.  104,  be  the  inacces- 
sible line,  and  G  the  point  from  which  it  is  desired  to  let  fall  the 
perpendicular  to  AB.  Through  G  run  out  and  measure  a  line 
of  any  convenient  length,  as  CD,  and  measure  the  angles  ACB 


PLANE    SURVEYING  153 

DCB,  and  DCA.  Set  up  the  instrument  at  D,  and  measure  the 
angles  ADC  and  BDC.  In  the  triangle  BDC,  we  have  given  two 
angles  and  the  included  side,  from  which  can  be  calculated  the 
length  of  the  side  CB,  In  the  triangle  ADC,  we  have  given  two 
angles  and  the  included  side,  from  which  can  be  calculated  the 
length  of  the  side  AC.  Then,  in  the  triangle  ACB,  we  have 
the  lengths  of  the  sides  AC  and  CB,  and  the  included  angle 
ACB,  from  which  can  be  calculated  the  angle  CAB.  If,  then,  the 
instrument  be  set  up  at  C,  and  an  angle  ACE  be  turned  off  equal 
to  90°  -  BAC,  the  line  of  sight 
will  point  in  the  direction  of  the  A 
required  perpendicular,  and  the 
length  of  the  perpendicular  will 
be  given  by  AC  cos  ACE. 

This  same  method  will  serve 
to  trace  a,  line  through  a  given 
point  parallel  to  an  inaccessi-  Fig.  104. 

l>le  line.    For  if,  with  the  instru- 
ment at  C,  an   angle    ACA'    be  turned  off  equal  to   CAB,    the 
line  A'  B'  will  be  parallel  to  AB. 

Obstacles  to  Alignment.  By  Perpendiculars:  "When  a  tree, 
house,  or  other  obstacle  obstructing  the  line  of  sight  (see  Fig.  105) 
is  encountered,  set  up  the  transit  at  the  point  B,  turn  off  a  right 
angle,  and  measure  the  length  of  the  line  BC.  Erect  a  second 
perpendicular  CD  at  C,  and  measure  its  length.  At  D  erect  a 
third  perpendicular  DE,  making  DE  =  BC.  Then  the  fourth 
perpendicular  EF  will  be  in  the  direction  of  the  required  line. 
The  distance  from  B  to  E  will  be  given  by  CD.  If  perpendic- 
ulars cannot  be  conveniently  set  off,  let  BC  and  DE  make  any 
equal  angle  with  the  line  AB,  so  that  CD  will  be  parallel  to  it. 

By  an  Equilateral  Triangle.  At  B  turn  off  from  the  direc- 
tion of  AB  produced,  an  angle  of  60°  in  the  direction  of  BC  (see 
Fig.  106),  and  make  BC  any  convenient  length  sufficient  to  clear 
the  obstacle.  Set  up  the  instrument  at  C  and  turn  off  an  angle  of 
60°  from  BC  to  CD  and  make  CD  of  a  length  equal  to  BC 
Finally  at  D  turn  off  a  third  angle  of  00°  from  CD  to  DB,  and 
the  line  DE  will  be  in  the  direction  of  AB  produced.-  The  dis- 
tance BD  will  equal  BC  or  CD. 


154  PLANE    SUKVEYING 


JJy  Tricmgulation.  Let  AB,  Fig.  107,  be  the  line  to  be 
prolonged  beyond  tlie  obstacle.  Choose  some  point  as  C  from 
which  can  be  seen  the  line  AB  as  well  as  some  point  D  beyond 
the  obstacle.  With  the  transit  at  A,  measure  the  length  of  AB 
and  measure  the  angle  BAG.  Set  up  the  transit  at  C,  and  meas- 


Fig.  105.  Fig.  106. 

ure  the  angles  BOA  and  ACD.  Then,  in  the  triangle  ACB, 
we  have  one  side  and  two  angles  known,  from  which  can  be  cal- 
culated the  lengths  of  AC  and  BC.  In  the  triangle  ACD,  we 
know  the  length  AC  and  the  angles  BAC  and  ACD,  from 
which  can  be  calculated  the  length  CE,  the  angle  BEC,  and  the 
distance  AE.  Therefore  from  C  measure  the  distance  CE,  set  up 
the  transit  at  E,  and  turn  off  the  angle  CEF  equal  to  180°  minus 
the  angle  BEC,  for  the  direction  of  the  required  line.  The 
length  of  BE  will  evidently  equal  AE  -  AB. 

Jjy  a  Random  L'ine.  When  a  wood,  hill,  or  other  obstacle 
prevents  one  end  of  a  line  (as  B,  Fig.  108)  being  seen  from  the 
other  end  A,  run  out  and  measure  a  random  line,  as  AC,  as 


Fig.  108. 

nearly  in  the  required  direction  as  may  be  guessed,  until  a  point 
C  is  reached  from  which  B  can  be  seen.  Xow,  if  convenient, 
measure  the  perpendicular  offset  from  AC  to  the  point  B,  from 

BC 
which    can  be  calculated   the  angle  CAB.     Tan.    CAB  =  — r-p . 

If  a  right  angle  cannot  be  turned  off  at  C,  turn  off  any  convenient 
angle  and  measure  the  distance  CB;  then,  in  the  triangle  ACB, 
there  are  given  two  sides  and  the  included  angle,  from  which  can 
be  calculated  the  anjj;le  CAB.  Kovv  taking  the  transit  back  to 
A,  the  angle  CAB  can  be  turned  off  in  the  proper  direction  from 


PLANE    SURVEYING  155 

AC,    and  the  correct  line    AB  can  be    run  out  and  measured  in 
the  proper  direction. 

By  Latitudes  and  Departure*.  When- a  single  line  such  as 
AC  cannot  be  run  so  as  to  come  opposite  the  given  point  B  (Fig. 
109),  a  series  of  zigzag  lines  (as  AC,  CD,  DE,  EF,  and  FB)  can 
be  run  in  any  convenient  direction,  so  as  at  last  to  arrive  at  the 
desired  point  B.  Any  one  of 
these  lines  (as,  for  instance,  AC) 
may  be  taken  as  a  meridian  to 
which  all  of  the  others  may  be 
referred,and  their  bearings  there- 
from deduced.  Calculate  the 
total  latitudes  and  departures  of  Fig.  109. 

these  lines,  as  AX  and  BX;  then 
the  bearing  of  the  required  line  BA  with  respect  to  AC  will  be 

T>V 

given  by  Tan.  BAG  =  -j^. 

By  Triangulation.  When  obstacles  prevent  the  use  of 
either  of  the  preceding  methods,  if  a  point  C  can  be  found  from 
which  A  and  B  are  accessible  (see  Fig.  110),  measure  the  dis- 
tances CA  and  CB,  and  the  angle  ACB,  from  which  can  be  calcu- 
lated the  length  of  the  side  AC  and  the  angle  CAB.  Now 

measure  the  angle  ACD  to  some 
point  D  beyond  the  obstacle; 
then,  in  the  triangle  ACD,  we 
have  two  angles  and  the  included 
side,  from  which  may  be  calcu- 
lated the  length  of  the  side  CD. 
Fig,  110.  Measure  the  distance  CD  in  the 

proper  direction,  set  up  the  tran- 
sit at  D,  and  turn  off  an  angle  CDB  equal  to  the  supplement  of 
ADC,  for  the  direction  of  the  required  line. 

The  distance  from  A  to  D  may  also  be  calculated  from  the 
triangle  ACD,  the  stake  at  D  given  its  proper  number,  and  the 
line  continued.  If  the  distances  CA  and  CB  cannot  be  meas- 
ured, it  will  be  necessary  to  measure  a  base-line  through  C,  from 
the  extremities  of  which  the  angles  to  A  and  B  can  be  measured 
and  the  required  distances  calculated  as  before. 


156  PLANE    SURVEYING 


The  following  problem,  as  illustrated  in  Fig.  Ill,  is  of  fre- 
quent occurrence  in  line  surveys.  The  line  AB  of  the  survey 
having  been  brought  up  to  one  side  of  a  stream,  it  is  desired  to 
continue  the  line  of  the  survey  across  the  stream  to  the  point  C, 
the  latter  point  being  visible  from  B  and  accessible.  It  is  required 
to  find  the  length  of  the  line  BC,  that  the  stake  at  0  may  be  given 
its  proper  number,  and  the  survey  continued  from  that  point. 
With  the  transit  at  B,  turn  off  the  required  angle  to  locate  the 
point  0,  and  drive  a  stake  at  that  point.  If  possible,  deflect  from 
BC  a  right  angle  to  some  point  E,  and  measure  the  length  of  BE. 
Take  the  transit  to  E,  and  measure  the  angle  BEC.  The  dis- 
tance BC  is  therefore: 

BC;  =  BE  tan  BEC. 

If  it  is  not  possible  to  turn  off  a  right  angle  at  B,  then  through 
B  run  a  line  (as  BE')  in  any  convenient  direction,  and  measure 
its  length;  measure  also  the  angles  E'BC  and  BE'C.  In  the 
triangle  CBE',  there  are  then  given  two  angles  and  the  included 

side,  from  which  the  side  BC  can 
be  calculated.  Should  it  be  nec- 
essary to  take  soundings  at  cer- 
tain intervals  (as,  say,  50  or  100 
feet  across  the  stream),  then  in 
the  triangle  BE  X  there  are  given 
the  distance  BX,  the  distance 
E'X,  and  the  angle  XBE',  from 
which  can  be  calculated  the  angle  BE'X.  With  the  transit  at  E', 
turn  off  from  BE'  the  angle  BE'X.  Now.  starting  a  boat  from 
the  shore,  direct  it  in  line  from  B  to  C  until  it  comes  upon  the 
line  of  sight  of  the  transit  from  E'  to  X.  At  that  point  take 
soundings,  and  similarly  for  the  point  X',  etc.  If  the  point  C  is 
not  visible  from  B,  find  some  point,  as  E  (see  Fig.  112),  from  which 
B  and  C  are  visible,  and  measure  the  angle  BEC  and  the  distance 
EB.  Find  a  second  point,  as  F,  from  which  E  and  C  are  visible, 
and  measure  the  angles  CEF  and  EFC  and  the  distance  EF.  Then, 
in  the  triangle  ECF,  there  are  given  two  angles  and  the  included 
side,  from  which  can  be  calculated  the  distance  EC.  In  the  tri- 
angle  BCE,  then,  there  are  given  two  sides  and  the  included 


PLANE    SURVEYING 


157 


angle,  and  from  these  the  third  side  BO  and  the  angle  EBC  can 
be  found.  The  stake  C  can  now  be  numbered,  and  the  bearing  of 
BC  deduced. 

EXAMPLES  FOR  PRACTICE. 

1.  In  Fig.  Ill,  given  BE'  =  210  feet;  angle  CBE'  =  110° 
15';    angle    BE'C  =  34°20';  stake  B  numbered  8  +  54.       It    is 
required  to  -find  the  number  of  the  stake  C. 

2.  (a)     In  Fig.  112,  given 
EF  =  250  feet;  BE  =  128  feet; 
angle  EFC  =  46°  40';  angle  CEF 

-  103°  30';  angle  BEG  =  39°  10'. 
If  the  stake  at  B  is  numbered  12 
+  20,  it  is  required  to  find  the 
number  of  the  stake  at  C. 

(5)    If  the  bearing  of  the  line 
AB  is  S  75°E,  and  the  deflection 

angle  of  BE  from  AB  is  104°  to  the  right,  find  the  bearing  of  BC. 
To  Supply  Omissions.     Any  two  omissions  in   a  closed  sur- 
vey— whether  of  the  direction  or  of  the  length,  or  of  both,  of  one 
or    more    lines    of  the    survey — can    always   be  supplied  by  the 

application  of  the  principle  of  lati- 
tude and  departures,  although  this 
method  should  be  resorted  to  only 
in  cases  of  absolute  necessity, 
since  any  omission  renders  the 
checking  of  the  field  work  im- 
possible. In  the  following  para- 
graphs, the  methods  outlined  will 
apply  equally  whether  the  survey 
has  been  made  with  the  transit 
or  with  the  compass. 

CASE  1.       When   the    length 
and  bearing  of  any  one  side  are 
wanting.     In   Fig.    113,  let  the 
Fig.  113.  dotted    line    FG    represent    the 

course  whose  length  and  bearing 
are  wanting.     Calculate  the  latitudes  and  departures  of  the  remain- 


158 


PLANE    SURVEYING 


ing  courses;  and  since  in  a  closed  survey  the  algebraic  sum  of  the 
latitudes  and  departures  should  equal  zero,  therefore  the  difference 
of  the  latitudes  will  be  the  latitude  of  the  missing  line,  and  the 
difference  of  the  longitudes  will  be  the  required  longitude.  The 

latitude  and  longitude  of  the  line, 
form  the  sides  of  a  right  triangle, 
from  which  we  have: 

Longitude 
Tangent  of  Bearing  = 


114. 


The  required  length  will  be  given 

Latitude 

by  L  =  7^  —  =R  —  :  —  . 
J  Cos  Bearing 

CASE  2.  When  the  length  of 
one  side  and  the  bearing  of  an- 
other are  wanting. 

(a)     WHEN    THE    DEFICIENT 

SIDES  ADJOIN  EACH  OTHER.      Ill  Fig. 

114,  let  the  bearing  of  DE,  and 
the  length  of  FE,  be  lacking. 
Draw  DF.  From  the  preceding 
proposition  we  can  calculate  the 
bearing  and  length  of  DF,  as 
though  DE  and  EF  did  not  exist. 
Then,  in  the  triangle  DEF,  we 
have  given  the  lengths  DF  and 
DE  and  the  angle  DEF,  from 
which  can  be  calculated  the  angle 
FDE  and  the  length  EF. 

(1}  WHEN  THE  DEFICIENT 
SIDES  AliE  SEPARATED  FROM  EACH 
OTHER.  Iii  Fig.  115  let  ABCDE 
FGA  represent  a  seven-sided 
field,  in  which  the  length  of  CD, 
and  the  bearing  of  FG,  are  want-  - 
ing.  Draw  DB',  B'A',  A'G',  of 
the  same  lengths,  and  parallel 
respectively  to  CB,  BA,  and  AG. 


115. 


Connect  G'  with  GE  and.F. 


Then,  in.  the  figure  DB'A'G'E,  there  are  given  the  lengths  and 


PLANE    SURVEYING 


159 


bearings  of  all  of  the  courses  but  G'E.  The  length  and  bearing  of 
the  last  course  can  be  calculated  by  the  principles  of  Case  1.  Then, 
in  the  triangle  EFG ,  there  are  given  the  lengths  and  bearings  of 
EF  and  EG',  from  which  can  be  calculated  the  length  and  bearing 
of  FG'.  Therefore,  in  the  triangle  GFG',  since  GG'  is  equal  in 
length  and  parallel  to  CD,  there  are  given  the  lengths  of  GF  and 
FG',  and  the  bearings  of  GG'  and  FG',  from  which  can  be  cal- 
culated the  length  of  GG'  and  the  bearing  of  GF. 

CASE  3.      When  thelengths  of  two  sides  arc  (ranting. 

(a)   WHEN  THE  DEFICIENT  SIDES  ADJOIN  EACH  OTHER.     In  the 
seven-sided  Fig.  116,  let  the  lengths  of  DE  and  EF  be  wanting. 
Calculate  the  length  and  bearing  of  DF  by  the  principles  of  Case  1. 
Then,  in  the  triangle .  EDF,  there 
are  given  the  angles  at  D  and  F, 
and  the  length  of  DF,  from  which 
can  be  calculated  the  lengths  of 
DE  and  EF. 

(i)      WHEN  THE  DEFICIENT 

SIDES    ARE    SEPARATED    FROM    EACH 

OTHER.  In  Fig.  115,  let  the 
lengths  of  CD  and  GF  be  want- 
ing. As  before,  having  calculated 
the  length  and  bearing  of  FG', 
in  the  triangle  FGG',  the  angle  at 
G  can  be  calculated  from  the  bear- 
ings of  FG  and  GG' ;  the  angle  at 
G'  from  the  bearings  of  GG'and  FG';  and  the  angle  at  F  from  the 
bearings  of  FG  and  FG'.  There  are  given  then  the  three  angles 
of  the  triangle,  and  the  length  of  one  side,  from  which  can  be 
calculated  the  lengths  of  the  other  sides. 

CASE  4.      When  the  hearings  of  two  sides  are  wanting. 

(«)  WHEN  THE  DEFICIENT  SIDES  ADJOIN  EACH  OTHER.  In  Fig. 
116,  find  the  length  and  bearing  of  DF  as  before.  Then,  in  the 
triangle  DEF,  there  are  given  the  lengths  of  the  three  sides,  from 
which  can  be  calculated  the  required  angles. 

(J)  WHEN  THE  DEFICIENT  SIDES  ARE  SEPARATED  FROM  EACH 
OTHER.  In  Fig.  115,  let  the  bearings  of  CD  and  GF  be  wanting. 
Calculate  the  length  and  bearing  of  FG'  as  before.  Then,  in  the 


\ 


Fig.  116. 


PLANE    SURVEYING 


triangle  FGG',  there  are  three  sides  known,  from  which  can  be  cal- 
culated the  three  angles,  and  therefore  the  bearings  can  be  deduced. 

UNITED  STATES  PUBLIC  LAND  SURVEYS. 

The  first  surveys  of  the  public  lands  of  the  United  States  were 
carried  out  in  Ohio,  under  an  act  of  Congress  approved  May  20th, 
1785.  This  act  provided  for  townships  6  miles  square,  containing 
36  sections  of  1  mile  square.  The  townships  6  miles  square,  were 
laid  out  in  ranges,  extending  northward  from  the  Ohio  River,  the 
townships  being  numbered  from  south  to  north,  and  the  ranges 
from  east  to  west.  The  territory  embraced  in  these  early  surveys 
forms  a  part  of  the  present  state  of  Ohio  and  is  known  as  "  The 
Seven  Ranges."  The  sections  were  numbered  from  1  to  36  com- 


W 


36 

30 

24 

16 

12 

6 

35 

29 

23 

17 

II 

5 

34 

28 

22 

16 

10 

4 

33 

27 

21 

15 

9 

3 

32 

26 

20 

14 

8 

2 

31 

25 

19 

13 

7 

1 

W 

6 

5 

4 

3 

2 

1 

7 

6 

9 

to 

II 

12 

id 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

Fig.  117. 


Fig.  118. 


mencing  with  No.  1  in  the  southeast  corner  of  the  township,  and 
running  from  south  to  north  in  each  tier,  to  No.  36  in  the  north- 
west corner  of  the  townships  as  shown  in  Fig.  117. 

A  subsequent  act  of  Congress,  approved  May  18th,  1796,  pro- 
vided for  the  appointment  of  a  surveyor  general,  and  directed  the 
survey  of  the  lands  northwest  of  the  Ohio  River,  and  above  the 
mouth  of  the  Kentucky  River.  This  act  provided  that  **  the  sec- 
tions shall  be  numbered  respectively,  beginning  with  the  number 
one  in  the  northeast  section,  and  proceeding  west  and  east  alter- 
nately, through  the  township,  with  progressive  numbers  till  the 
thirty-sixth  be  completed."  This  method  is  shown  in  Fig.  118 
and  is  still  in  use. 

An  act  of  Congress,  approved  Feb.  llth  1805,  directs  the  sub- 
division  of  the  public  lands  into  quarter  sections,  and  provides  that 


PLANE    SURVEYING  161 


all  the  corners  marked  in  the  public  surveys  shall  be  established 
as  the  proper  corners  of  sections,  or  subdivisions  of  sections,  which 
they  were  intended  to  designate,  and  that  corners  of  half  and 
quarter  sections  not  marked  shall  be  placed,  as  nearly  as  possible, 
"  equidistant  from  those  two  corners  which  stand  on  the  same  line." 
This  act  further  provides  that  '•  the  boundary  lines  actually  run 
and  marked  *  *  *  shall  be  established  as  the  proper  bound- 
ary lines  of  the  sections  or  subdivisions  for  which  they  were 
intended;  and  the  length  of  such  lines  as  returned  by  *  *  * 
the  surveyors  *  *  *  shall  be  held  and  considered  as  the 
true  length  thereof  " 

An  act  of  Congress,  approved  April  24th,  1820,  provides  for 
the  sale  of  public  lands  in  half-quarter  sections,  and  requires  that 
"  in  every  case  of  the  division  of  a  quarter  section  the  line  for  the 
division  thereof  shall  run  north  and  south.  An  act  of  Congress, 
approved  April  5th,  1832,  directed  the  subdivision  of  the  public 
lands  into  quarter -quarter-sections  and  that  in  every  case  of  the 
division  of  a  half-quarter  section,  the  dividing  line  should  run 
east  and  west;  and  that  fractional  sections  should  be  subdivided 
under  rules  and  regulations  prescribed  by  the  Secretary  of  the 
Treasury. 

By  an  act  of  Congress,  approved  March  3rd,  1849,  the  Depart- 
ment of  the  Interior  was  created,  and  the  act  provided  "That the 
Secretary  of  the  Interior  shall  perform  all  the  duties  in  relation  to 
the  General  Land  Office,  of  supervision  and  appeal  now  discharged 
by  the  Secretary  of  the  Treasury.  *  *  *"  By  this  act  the 
General  Land  Office  was  transferred  to  the  Department  of  the 
Interior  where  it  still  remains. 

The  division  of  the  public  lands  is  effected  by  means  of  merid- 
ian lines  and  parallels  of  latitudes  established  six  miles  apart. 
The  squares  thus  formed  are  called  Towns/tips,  and  contain  36 
square  miles,  or  23,040  acres  "  as  nearly  as  may  be.  "  All  of  the 
townships  situated  north  or  south  of  each  other,  form  a  Range  and 
are  named  by  their  number  east  or  west  of  the  principal  meridian. 
Thus,  the  first  range  west  of  the  meridian  would  be  designated  as 
Range  1  West  ( E.  1.  W.).  Each  tier  of  townships  is  named  by 
its  number  north  or  south  of  the  base  line,  as  Township  2  North 
(T.  2.  N.). 


162  PLANE    SURVEYING 

Existing  laws  further  require  that  each  township  shall  be 
divided  into  thirty-six  sections,  by  two  sets  of  parallel  lines,  one 
governed  by  true  meridians  and  the  other  by  parallels  of  latitude, 
the  latter  intersecting  the  former  at  right  angles,  at  intervals  of 
one  mile  ;  and  each  of  these  sections  must  contain,  as  nearly  as 
possible,  six  hundred  and  forty  acres.  These  requirements  are 
evidently  inconsistent  because  of  the  convergency  of  the  meridians, 
and  the  discrepancies  will  be  greater  as  the  latitude  increases. 

In  view  of  these  facts,  it  was  provided  in  section  3  of  the  act 
of  Congress  approved  May  10th,  1800,  that  "  in  all  cases  where  the 
exterior  lines  of  the  townships,  thus  to  be  subdivided  into  sections 
and  half-sections,  shall  exceed,  or  shall  not  extend  six  miles,  the 
excess  or  deficiency  shall  be  specially  noted,  and  added  to  or 
deducted  from  the  western  or  northern  ranges  of  sections  or  half- 
sections  in  such  township,  according  as  the  error  may  be  in  running 
lines  from  east  to  west,  or  from  south  to  north  ;  the  sections  and 
half-sections  bounded  on  the  northern  and  western  lines  of  such 
townships  shall  be  sold  as  containing  only  the  quantity  expressed 
in  the  returns  and  plots,  respectively,  and  all  others  as  containing 
the  complete  legal  quantity." 

To  harmonize  these  various  requirements  as  fully  as  possible, 
the  following  methods  have  been  adopted  by  the  general  land  office. 

Initial  points  are  first  establis  hed  astronomically  under  special 
instructions,  and  from  this  initial  point  a  "principal  meridian  "  is 
laid  out  north  and  south.  Through  this  initial  point  a  "base 
line"  is  laid  out  as  a  parallel  of  latitude  running  east  and  west. 
On  the  principal  meridian  and  base  lines,  the  half-mile,  mile  and 
six-mile  corners  are  permanently  located,  and  in  addition,  the 
meander  corners  at  the  intersection  of  the  line  with  all  streams, 
lakes  or  bayous  prescribed  to  be  meandered.  These  lines  may  be 
run  with  solar  instruments,  but  their  correctness  should  be  checked 
by  observations  with  the  transit  upon  Polaris  at  elongation. 

Standard  parallels,  also  called  correction  lines,  are  run  east 
and  west  from  the  principal  meridian  at  intervals  of  twenty-four 
miles  north  and  south  of  the  base  line,  and  the  law  provides  that 
"  where  standard  parallels  have  been  placed  at  intervals  of  thirty 
or  thirty-six  miles,  regardless  of  existing  instructions,  and  where 
gross  irregularities  require  additional  standard  lines,  from  which  to 


TOPOGRAPHER  OF  U.  S.  GEOLOGICAL  SURVEY  AT  WORK  ON  THE  PLAINS  OF  COLORADO 


PLANE    SURVEYING  163 


initiate  new,  or  upon  which  to  close  old  surveys,  an  intermediate 
correction  line  should  be  established  to  which  a  local  name  may  be 
given:  and  the  same  will  be  run,  in  all  respects,  like  the  regular 
standard  parallels." 

Guide  meridians  are  extended  north  from  the  base  line,  or 
standard  parallels,  at  intervals  of  twenty-four  miles  east  and  west 
of  the  principal  meridian. 

When  conditions  are  such  as  to  require  the  guide  meridians 
to  run  south  from  a  standard  parallel  or  a  correction  line,  they  are 
initiated  at  properly  established  closing  corners  of  the  given  paral- 
lel. That  is  to  say,  they  are  begun  from  the  point  on  the  parallel 
at  which  they  would  have  met  it  if  they  had  been  run  north  from 
the  next  southern  parallel.  This  point  is  obtained  from  computa- 
tion, and  is  less  than  twenty-four  miles  from  the  next  eastern  or 
western  meridian  by  the  convergence  of  the  meridians  in  twenty- 
four  miles. 

In  case  guide  meridians  have  been  improperly  located  too  far 
apart,  auxiliary  meridians  may  be  run  from  standard  corners,  and 
these  may  be  designated  by  a  local  name. 

The  angular  convergence  of  two  meridians  is  given  by  the 
equation 

(f>  =  m  sin  L  (1) 

where  m  is  the  angular  difference  in  longitude  of  the  meridians, 
and  L  is  the  mean  latitude  of  the  north  and  south  length  under 
consideration. 

The  linear  convergence  in  a  given  length  I  is 

c  =  I  sin  4>  (2) 

The  radius  of  a  parallel  at  any  latitude  L  is  given  by  the 
equation 

r  =  cos  L  (3) 

where  R  is  the  mean  radius  of  curvature  of  the  earth. 

The  distance  between  meridians  is  usually  given  in  miles  and 
this  must  be  reduced  to  degrees.  To  do  this  it  is  first  necessary  to 
find  the  linear  value  of  one  degree  of  longitude  at  the  mean  latitude 
from  the  proportion. 

1°  :  360°  :  :  a> :  2nr  (4) 

the  value  of  •/•  being  found  from  (3) 


164  PLANE    SURVEYING 


Equation  (4)  will  give  results  sufficiently  accurate,  although 
in  strict  accuracy  R  should  be  the  radius  of  curvature  at  the  mean 
latitude. 

For  full  details  of  public-land  surveying,  see  "  Manual  of 
Surveying  Instructions  for  the  Survey  of  the  Public  Lands  of  the 
United  States,"  issued  by  the  Commissioner  of  the  General 
Land  Office.  These  "  Instructions  "  are  prepared  for  the  direction 
of  those  engaged  on  the  public  land  surveys,  and  new  editions  are 
issued  from  time  to  time. 

Much  of  the  foregoing  in  very  condensed  form  is  taken  from 
the  edition  of  1894. 

The  following  table  gives  the  convergency  both  in  angular 
units  and  linear  units  for  township  6  miles  square,  between  lati- 
tudes 30°  and  70°  north: 

Let  it  be  required  to  find  from  the  table  the  linear  converg- 
ence for  a  township  situated  in  latitude  38°  29'  north. 

Looking  in  the  table  opposite  39°  we  find  the  linear  con- 
vergence. 

For  39°  =  58.8  links 

For  38°  =  56.8  links 

Difference  for  1°  =  2.0  links 
Difference  for  1'  =  2.0  -^  60  =  .0333  links 
Difference  for  29'  =  .0333  X  29  =  .97  links 

Therefore  total  convergence  for  latitude  38°  29'  =  56.8  4 
0.97  links  =  57.77  links. 


172 


PLANE    SURVEYING 


165 


BASE  HEASUREHENT. 

It  is  not  intended  in  what  follows  to  go  into  the  details  of  the 
measurement  of  a  base  for  an  extended  system  of  triangulation,  as 
that  properly  belongs  to  Geodetic  Surveying.  Some  description 
of  base  measuring  apparatus  will  be  given,  with  illustrations  of 


Lat. 

Convergency. 

Lat. 

Convergency 

On  the 
Parallel. 

Angle. 

On  the 
Parallel. 

An 

gle. 

Degrees. 

Links.         Minutes. 

Seconds. 

Degrees.       Links. 

Minutes. 

Seconds. 

30 

41.9 

3 

0 

50              86.4 

6 

12 

31 

43.G 

3 

7 

51              896 

6 

25 

32 

45.4 

3 

15 

52 

92.8 

6 

39 

33 

47.2      !        3 

23 

53 

96.2 

(5 

54 

34 

49.1              3 

30 

54 

99.8 

9 

35 

50.9             3 

38 

55 

103.5 

7 

25 

36              52.7              3 

46 

56 

107.5 

7 

42 

37              54.7 

3 

55 

57 

111.0 

8 

0 

38 

56.8 

4                04              58           116.0 

8 

19 

39 

58.8 

4                13              59           120.6 

8 

38 

40 

60.9 

4                22              60           125.5 

8 

59 

41 

63.1 

4                31              61            130.8 

!) 

22 

42 

65.4 

4                41              62           136.3 

<) 

46 

43 

67.7 

4                51              63           142.2 

10 

11 

44 

70.1 

5                   1               64            148.6 

10 

38 

45 

72.6 

5 

12              65 

155.0 

11 

8 

46 

75.2 

5 

23 

66           162.8 

11 

39 

47 

77.8 

5 

23 

67           170.7 

12 

13 

48              80.6 

5 

46 

68           179.3 

12 

51 

49              83.5 

5 

59 

69 

188.7 

13 

31 

70 

199.1 

14 

15 

various  devices,  and  special  attention  will  be  given  to  the  use  of 
the  tape  in  the  accurate  measurement  of  lines  such  as  occur  in  usual 
field  operations  of  Plane  Surveying. 

Much  of  what  follows  is  from  the  excellent  treatise  on  Topo- 
graphic Surveying  by  Herbert  M.  Wilson. 

A  trigonometric  survey  is  usually  carried  over  a  country  where 
the  direct  measurement  of  distances  is  impracticable,  and  since  the 
calculations  of  these  distances  proceeds  from  the  direct  measure- 
ment of  the  base-line,  this  base  line  should  be  so  located  as  to 
permit  of  its  length  being  determined  with  any  degree  of  accuracy 
consistent  with  the  nature  of  the  work  involved. 

To  attain  the  desired  results,  the  site  should  be  reasonably 
level  and  afford  room  for  a  base  of  proper  length  so  that  its  ends 
may  be  intervisible.  and  permit  of  the  development  of  a  scheme  of 


166  PLANE    SURVEYING 

primary  triangulation  giving  the  best-conditioned  figures  possible. 
Other  things  being  equal,  that  site  is  best  that  includes  solid 
ground;  both  for  permanency  of  monuments  and  facility  and 
accuracy  of  measurement. 

Base  Apparatus.  In  early  days,  base-lines  were  measured  by 
means  of  wooden  rods,  varnished  and  tipped  with  metal.  The  rods 
were  supported  in  trestles,  the  contacts  between  the  ends  being 
made  with  great  care.  Later,  compensated  rods  were  employed,  as 
for  instance  the  Contact-Slide  Apparatus  of  the  U.  S.  Coast  Survey 
and  the  Repsold  primary  base  bars  of  the  U.  S.  Lake  Survey,  see 
Fig.  119,  resulting  in  greater  accuracy  in  the  measurement  of  base 
lines.  The  use  of  the  iced  bar  (see  Fig.  120)  by  the  U.  S.  Coast 
Survey,  represents  the  highest  development  of  base-measuring 
apparatus. 


Fig.    119. 

Within  recent  years  the  steel  tape  has  become  popular  as  the 
accuracy  attainable  with  its  use  has  become  more  fully  appreciated. 
Errors  in  Base  Heasurement.      The  following  are  the  chief 
sources  of  error  in  base  measurement: 

1.  Changes  of  temperature  ; 

2.  Difficulties  of  making  contact ; 

3.  Variations  of  the  bars  or  tape  from  the  standards. 

The  refinements  of  measurement  consist  especially  in  — 

a.  Standardizing  the  measuring  apparatus,  or  its  comparison  with  a 
standard  of  lengtn. 

b.  Determination  of  temperature,  or  its  neutralization  by  the  use  of 
compensating  bars. 

c.  Means  adopted  for  reducing  the  number  of  contacts  to  the  fewest 
possible,  and  of  making  these  with  the  greatest  degree  of  precision. 


PLANE    SURVEYING 


167 


The  inherent  difficulties  of  measurement  with  lar*  of  any 
kind  are  : 

1.  Necessity  of  measuring  short  bases  because  of  the  number  of  times 
which  the  bar  must  be  moved. 

2.  Expense,  as  a  considerable  number  of  men  are  required. 

3.  Slowness,  the  measurement  often  occupying  from  a  month  to  six 
weeks. 

The  advantages  of  measurement  made  with  a  steel  tape  are  : 

1.  Great  reduction  in  the  number  of  contacts,  as  the  tapes  are  about 
three  hundred  feet  long  as  compared  with  bars  of  about  twelve  feet. 

2.  Comparatively  small  cost  because  of  the  few  persons  required. 

3.  Shortness  of  the  time  employed,  an  hour  to  a  mile  being  an  ordinary 
record  in  actual  measurement 

4.  Errors  in  trigonometric  expansion  may  be  reduced  by  increasing  the 
length  of  the  base  from  5  miles,  the  average  length  of  a  bar-measured  base, 
to  8  miles,  not  an  uncommon  length  for  tape-measured  bases. 


Fig.  120. 

Steel  tapes  offer  a  means  of  measuring  base  lines  which  is 
superior  to  that  obtained  by  measuring  bars,  because  they  combine 
the  advantages  of  great  length  and  simplicity  of  manipulation, 
with  the  precision  of  the  shorter  laboratory  standards,  providing 
only  that  means  be  perfected  for  eliminating  the  errors  of  tem- 
perature and  of  sag  in  the  tape.  Base  lines  can  be  so  conveniently^ 
and  rapidly  measured  with  long  steel  tapes  as  to  permit  of  their 
being  made  of  greater  length  than  has  been  the  practice  with  lines 
measured  by  bars,  and  as  a  result,  still  greater  errors  may  be 
introduced  in  tape-measured  bases  and  yet  not  affect  the  ultimate 


168 


PLANE    SURVEYING 


expansion  any  more  than  will  the  errors  in  the  latter,  because  of 
the  greater  length  of  the  base. 

The  tapes  used  for  this  work  are  of  steel,  either  300  feet  or 
100  meters  in  length.  The  tapes  used  by  the  Coast  Survey  are 
101.01  meters  in  length,  6.34  millimeters  by  0.47  millimeters  in 
cross -section,  and  weigh  22.3  grams  per  meter  of  length.  They  are 

subdivided  into  20  meter  spaces 
by  graduations  ruled  on  the  sur- 
face of  the  tape,  and  their  ends 
terminate  in  loops  obtained 
either  by  turning  back  and  an- 
nealing the  tape  on  itself,  or  by 
fastening  them  into  brass  hand- 
les. When  not  in  use,  the  tapes 
are  rolled  on  reels  for  easy  trans- 
portation. 

The  steel  tapes  used  by  the 
Geological  Survey  are  similar  to 
those  used  by  the  Coast  Survey, 
excepting  in  their  length,  whicb 
is  a  little  over  300  feet.  They 
are  graduated  for  300  feet  and 
are  subdivided  every  10  feet,  the 
last  5  feet  of  which  at  either  end 
is  subdivided  to  feet  and  tenths. 
The  various  instrument-makers  now  carry  such  tapes  in  stock, 
wound  on  hand -reels.  All  tapes  must  be  standardized  before  and 
after  use,  by  comparison  with  laboratory  standards,  and,  if  possible, 
thereafter  frequently  in  the  field  by  means  of  an  iced-bar  apparatus. 
In  measuring  with  steel  tapes,  a  uniform  tension  must  be 
applied.  In  order  to  get  a  uniform  tension  of  20  to  25  pounds, 
some  form  of  stretcher  should  be  used.  That  used  by  the  U.S. 
Coast  Survey  consists  of  a  base  of  brass  or  wood,  2  or  3  feet 
in  length  by  a  foot  in  width,  upon  which  is  an  upright  metallic 
standard,  and  to  this  is  attached  by  a  universal  joint,  an  ordinary 
spring-balance,  to  which  the  handle  of  the  tape  is  fastened.  See 
Fig.  121.  The  upright  standard  is  hinged  at  its  junction  with 
the  base,  so  that  when  the  tape  is  being  stretched,  the  tapeman 


Fig.  121. 


PLANE    SURVEYING 


(T 


1 0 


?! 
^ 


~ 

t 


-•     -o 

x    a 
x  s 

M 

^    be 

^    c 

ts.  -r 


17o  PLANE    SURVEYING 

can  put  the  proper  tension  on  it  by  taking  hold  of  the  upper  end 
of  the  upright  standard  and  using  it  as  a  lever,  and  by  pulling  it 
back  toward  himself  he  is  enabled  to  use  a  delicate  leverage  on  the 
balance  and  attain  the  proper  pull. 

The  thermometers  used  are  ordinary  glass  thermometers, 
around  the  bubbles  of  which  should  be  coiled  thin  annealed  steel 
wire,  so  that  by  passing  them  in  the  air  adjacent  to  the  tape,  a 
temperature  corresponding  to  that  of  the  tape  can  be  obtained. 
Experience  with  such  thermometers  shows  that  they  closely  fol- 
low the  temperature  of  the  steel  tape.  For 
the  best  results,  two  thermometers  should 
be  used,  each  at  about  one-fourth  of  the 
distance  from  the  extremities  of  the  tape. 
The  stretching  device  used  by  the  U.  S. 
Geological  Survey  is  much  simpler  and 
more  quickly  manipulated  than  that  of  the 

1  Coast  Survey.     The  chief  object  to  be  at- 

— I  tained   in  tension  is  steadiness  and  uni- 
Fig.  123.  formity  of  tension;   the    simplest  device 

which  will  attain  this  end  is  naturally  the 

best.  Two  general  forma  of  such  devices  are  employed  by  the  U.  S. 
Geological  Survey,  one  for  the  measurement  of  base  lines  along 
railways,  where  the  surface  of  the  ties  or  the  roadbed  furnishes 
support  for  the  tape,  and  the  device  must  therefore  be  of  such 
kind  as  to  permit  of  the  ends  being  brought  close  to  the  surface; 
the  other  is  employed  in  measurements  made  over  rough  ground, 
where  the  tape  may  frequently  be  raised  to  considerable  heights 
above  the  surface  and  be  supported  upon  pegs. 

The  stretcher  used  by  the  Geological  Survey  for  measuring 
on  railways  is  illustrated  in  Fig.  122,  and  was  devised  by  Mr.  He 
L.  Baldwin.  It  consists  of  an  ordinary  spring-balance  attached  to. 
the  forward  end  of  the  tape,  where  a  tension  of  twenty  pounds  is 
applied,  the  rear  end  of  the  tape  being  caught  over  a  hook  which 
is  held  steadily  by  a  long  screw  with  a  wing-nut,  by  which  the 
zero  of  the  tape  may  be  exactly  adjusted  over  the  mark  scratched 
on  the  zinc  plate.  The  spring-balance  is  held  by  a  wire  running 
over  a  wheel,  which  latter  is  worked  by  a  lever  and  held  by 
ratchets  in  any  desired  position,  so  that  by  turning  the  wheel,  a 


PLANE    SURVEYING  171 

uniform  strain  is  placed  on  the  spring-balance,  which  is  held  at 
the  desired  tension  by  the  ratchets. 

The  tape- stretcher  used  by  the  U.  S.  Geological  Survey  off 
railways  consists  of  a  board  about  5  feet  long,  to  the  forward  end 
of  which  is  attached  by  a  strong  hinge,  a  \vooden  lever  about  5 
feet  in  length,  through  the  larger  portion  of  the  length  of  which 
is  a  slot.  See  Fig.  123.  Through  the  slot  is  a  bolt  with  wing- 
nut,  which  can  be  raised  or  lowered  to  an  elevation  corresponding 
with  the  top  of  the  hub  over  which  measurement  is  being  made; 
hung  from  the  bolt  is  the  spring- balance,  to  which  the  forward 
tapeman  gives  the  proper  tension  by  a  direct  pull  on  the  lever, 
the  weight  of  the  lever  and  the  friction  in  the  hinge  being  such  as 
to  make  it  possible  to  bring  about  a  uniform  tension  without  dif- 
ficulty. The  zero  on  the  rear  end  of  the  tape  is  adjusted  over  the 
contact  mark  on  the  zinc  by  means  of  a  similar  lever  with  hook- 
bolt  and  wing-nut,  but  without  the  use  of  spring-balance. 

LAYING  OUT  THE  BASE.  The  most  laborious  operation  in  base 
measurement  is  its  preliminary  preparation,  which  consists  of: 

1.  Aligning  with  transit  or  theodolite; 

2.  Careful  preliminary  measurement  for  the  placing  of  stakes  on    rough 

ground; 

3.  Placing  of  zinc  marking-strips  on  the  stakes. 

Base  lines  measured  with  steel  tapes  across  country  are  aligned 
with  transit  or  theodolite,  and  are  laid  out  by  driving  large  hubs 
of  3  X  6  scantling  into  the  ground,  the  tops  of  the  same  project- 
ing to  such  a  height  as  will  permit  a  tape-length  to  swing  free  of 
obstructions.  These  large  hubs  are  placed  by  careful  preliminary 
measurement  at  exact  tape-lengths  apart,  and  between  them  as  sup- 
ports, long  stakes  are  driven  at  least  every  50  feet.  Into  the  sides 
of  these  near  their  tops  are  driven  horizontally,  long  nails,  which 
are  placed  at  the  same  level  by  eye,  by  sighting  from  one  terminal 
hub  to  the  next.  The  tape  rests  on  these  nails  and  on  the  surface 
of  the  terminal  hubs  are  tacked  strips  of  zinc  on  which  to  make  the 
contact  marks.  A  careful  line  of  spirit-levels  must  be  run  over 
the  base-lines,  and  the  elevation  of  the  hub  or  contact-mark  of  each 
tape-length  must  be  determined  in  order  to  furnish  data  for 
reduction  to  the  horizontal. 

In  measuring  over  rough  ground,  six  men  are  necessary:  two 
tape -stretchers,  two  markers,  two  observers  of  thermometers,  one 


172  PLANE    SURVEYING 

of  whom  will  record.  The  co-operation  of  these  men  is  obtained 
by  a  code  of  signals,  the  first  of  which  calls  for  the  application  of 
the  tension;  then  the  two  tape- stretchers  by  signal  announce  when 
the  proper  tension  has  been  applied;  then  the  rear  observer  ad  justs 
the  rear  graduation  over  the  determining  mark  on  the  zinc  plate 
and  gives  a  signal,  upon  hearing  which,  the  thermometer  recorder 
near  the  middle  of  the  tape  lifts  it  a  little  and  lets  it  fall  on  its 
supports,  thus  straightening  the  tape.  Immediately  thereafter  the 
front  observer  marks  the  position  of  the  tape  graduation  on  the 
zinc  plate,  and  at  the  same  time  the  thermometers  are  read  and 
recorded. 

After  the  measurement  of  the  base  line  has  been  completed  in 
the  field,  the  results  of  the  measurement  have  to  be  reduced  for 
various  corrections,  among  which  are: 

Comparison  with  standard  measure: 

Corrections  for  inclination  and  sag  of  tape  if  such  is  used; 

Correction  for  temperature. 

The  first  correction  to  be  applied  is  that  of  reducing  the  tape- 
line  to  the  standard,  "  standardizing  "  the  tape  as  it  is  called.  By 
sending  the  tape  to  the  National  Bureau  of  Standards  at  Washing- 
ton, D.  C.,  a  statement  may  be  had  of  the  length  of  the  tape  com- 
pared with  the  standard.  For  this  service  a  small  fee  is  charged. 
For  an  additional  fee  a  statement  may  be  had  of  the  temperature 
and  pull  at  the  ends  for  which  the  tape  is  a  standard. 

As  the  length  of  a  steel  tape  varies  with  the  temperature,  one 
of  the  most  uncertain  elements  in  the  measurement  of  a  base  with 
the  steel  tape,  is  the  change  in  the  length  of  the  standard  due  to 
changes  of  temperature.  Corrections,  therefore,  must  be  made  for 
every  tape-length  as  derived  from  readings  of  one  or  more  ther- 
mometers applied  to  the  tape  in  the  course  of  measurement. 

Steel  expands  .0000063596  of  its  length  for  each  degree 
Fahrenheit.  This  decimal  multiplied  by  the  average  number  of 
degrees  of  temperature  above  or  below  62  degrees  at  the  time  of 
the  measurement,  gives  the  proportion  by  which  the  base  is  to  be 
diminished  or  extended  on  account  of  temperature  changes.  This 
correction  is  applied  usually  by  obtaining  with  great  care,  the 
mean  of  all  thermometer  readings  taken  at  uniform  intervals  of 
distance  during  the  measurement. 


PLANE    SURVEYING  173 

The  data  for  the  correction  for  inclination  of  base  are  obtained 
by  a  careful  line  of  spirit-levels  over  the  base-line.  In  the  course 
of  this  leveling,  elevations  are  obtained  for  every  plug  upon  which 
the  tape  rests.  The  result  of  this  leveling  is  to  give  a  profile 
showing  rise  or  fall  in  feet  or  fractions  thereof  between  the  points 
of  change  in  inclination  of  the  tape-line.  From  this  and  measured 
distances  between  these  points,  the  angle  of  inclination  is  com- 
puted by  the  formula 


In  which  D  is  the  length  of  the  tape  or  measured  base  : 
and  //  is  the  difference  in  height  of  the  ends  of  tape  or 
measured  base,  expressed  in  feet. 
0  is  the  angle  of  slope  expressed  in  minutes. 

The  correction  in   feet   to   the  distance   is  that  computed  by  the 
equation 

Correction  =  D  8in'     *'  ffi 

An  approximate  formula  for  reducing  distances  measured 
upon  sloping  ground  to  the  horizontal  is  expressed  by  the  rule  : 
Divide  the  square  of  the  difference  of  level  by  twice  the  measured 
distance,  subtract  the  quotient  thus  found  from  the  measured 
distance,  and  the  remainder  equals  the  distance  required  ;  thus 


in  which  d  equals  the  horizontal  or  reduced  distance. 

When  the  base  measurement  is  made  with  steel  tape  across 
country,  and  accordingly  is  not  supported  in  every  part  of  its 
length,  there  will  occur  some  change  in  its  length,  due  to  sag.  As 
previously  explained,  the  tape  should  be  rested  upon  supports  not 
less  than  50  feet  apart.  With  supports  placed  even  this  short  dis- 
tance apart,  however,  a  change  of  length  will  occur  between  them, 
while  even  greater  changes  will  occur  should  one  or  more  supports 
be  omitted  as  in  crossing  a  road,  ravine,  etc.  Since  tapes  are 
standardized  by  laying  them  upon  a  flat  standard,  it  is  necessary 
to  determine  the  amount  of  shortening  from  the  above  causes. 


174  PLANE    SURVEYING 

The  following  reduction  formulae  apply  : 

Let  w  =  weight  per  unit  of  length  of  tape  : 
t  =  tension  applied 
w 

a  =  — 

n  =  number  of  sections  into  which  tape  is  divided  by 

supports. 

I  —  length  of  any  section 

L  —  normal  length  of  tape  or  right-line  distance  be- 
tween n  marks  when  under  tension  :  =  nl  ap- 
proximately. 

If  a  tape  be  divided  by  equidistant  supports,  the  difference  in 
distance  between  the  end  graduations,  due  to  sag,  or  the  correction 
for  sag  =  dL  becomes 


If  one  or  more  supports  are  omitted,  then  the  omission  of  m 
consecutive  supports  shortens  the  tape  by 

J__m(m  +  1)  (m  +  2)  a2/3: 

when  I  is  the  length  of  the  section  when  no  supports  are  omitted. 
Example.     Let  n  =  6  ;  I   ==  50  feet  ;  w  —  .0145  =  weight 
in  pounds  per   foot  found   by  dividing  whole  weight  of  tape  by 
whole  length  ;  t  =  20  pounds. 


,/  =  —        =  0.0162  feet, 

24     V   t    ' 

which  is  the  amount  of  shortening  of  each  tape-length.     This  cor- 
rection is  always  negative. 

If  there  had  been  86  full  tape-lengths  in  measured  base-line, 
the  total  corrections  for  sag  would  be  86  X  .0162  =  1.393  feet. 

THE  PLANE-TABLE. 

Construction.  The  plane-table  consists  essentially  of  a  draw- 
ing-board mounted  upon  a  tripod.  This  board  is  usually  twenty- 
four  by  thirty  inches,  constructed  in  sections  to  prevent  warping; 
it  is  attached  to  the  tripod  by  a  three-screw  leveling  base  arranged 


PLANE    SURVEYING  176 


to  permit  the  board  to  be  turned  in  azimuth  and  to  be  clamped  in 
any  position. 

The  instrument  is  designed  to  at  once  sketch  in  the  field,  to 
scale,  the  lengths  and  relative  directions  of  all  lines  and  the  posi- 
tions of  objects  to  be  included  in  the  survey.  For  drawing 
straight  lines,  a  steel  ruler  is  provided  upon  which  is  mounted  at 
each  end,  a  pair  of  open  sights  like  those  of  the  compass,  or,  a  tele- 
scope is  mounted  at  the  center  of  the  ruler,  fitted  with  stadia 
wires,  a  vertical  arc  and  a  longitudinal  striding  level.  The  eye. 
piece  should  be  inverting,  and  whether  the  open  sights  or  the  tele- 
scope is  used,  the  line  of  sight  should  always  be  parallel  to  the 
edge  of  the  ruler.  The  straight  edge  with  the  attached  telescope 
or  open  sights  is  called  the  alidade. 

For  leveling  the  instrument,  two  cylindrical  levels,  at  right 
angles  to  each  other,  are  mounted  upon  the  alidade  and  either  an 
attached  or  detached  compass  is  provided  for  determining  the  bear- 
ing of  lines. 

For  attaching  the  paper  to  the  board,  various  devices  are 
used.  One  consists  of  a  roller  at  each  end  of  the  table  upon  one 
of  which  the  paper  is  wound  up  as  it  is  unrolled  from  the  other, 
the  edges  of  the  paper  being  held  close  to  the  board  by  spring 
clips.  This  arrangement  permits  the  paper  to  be  used  in  a  con- 
tinuous roll  and  to  be  tightly  stretched  over  the  board.  The  use 
of  the  continuous  roll  of  paper  is  undesirable,  however,  and 
separate  sheets  of  proper  size  should  be  used,  attached  to  the  board 
and  held  firmly  in  place  by  the  spring  clips  provided  with  the 
instrument.  The  use  of  thumb-tacks  should  be  avoided. 

Under  the  most  favorable  conditions,  the  plane-table  is  a  very 
awkward  instrument  and  difficult  to  handle,  but  it  is  admirably 
adapted  to  filling  in  the  details  of  a  topographical  survey.  For  this 
purpose  it  is  the  standard  instrument  of  the  United  States  Geo- 
detic Survey  and  is  also  largely  used  by  the  United  States  Geolog- 
ical Survey.  It  cannot  be  used  on  damp  or  very  windy  days  and  is 
not  therefore,  of  as  general  utility  as  the  transit  and  stadia. 

Fig.  124  shows  one  form  of  construction  of  the  plane  table 
with  leveling  screws  and  Fig.  124«  shows  a  plane  table  with  a  much 
simpler  form  of  leveling  head.  This  latter  was  designed  by  Mr. 
W.  D.  Johnson  and  has  received  the  approval  of  the  topographers 


176 


PLANE    SUKVEYING 


of  the  United  States  Geological  Survey.  The  whole  arrangement 
is  very  light,  but  does  not  permit  of  as  close  leveling  as  does  the 
usual  form  with  leveling  screws. 

Adjustments. 

1st.      To  determine  whether  the  edge  of  the  ruler  is  straight. 


Fig.  124. 

Place  the  ruler  upon  a  smooth  surface,  and  draw  a  line  along  its 
edge,  and  also  lines  at  its  ends.  Reverse  the  ruler  on  these  lines, 
and  draw  another  line  along  its  edge.  If  these  two  lines  coincide, 
the  ruler  is  straight. 


PLANE    SURVEYING 


177 


2nd.  To  make  the  plane  of  the  table  horizontal  when  the 
bubbles  are  in  the  center  of  the  tubes.  Assuming  the  table  to  be 
plane,  set  the  alidade  in  the  middle  of  the  table,  level  by  means  of 
the  leveling  screws,  draw  lines  along  the  edge  and  ends  of  the 
ruler,  and  reverse  the  alidade  on  these  lines.  If  the  bubbles 
remain  in  the  center  of  the  tubes,  they  are  in  adjustment.  If  they 


Fig.  124a. 

do  not,  correct  one-half  of  the  error  by  means  of  the  leveling 
screws  and  the  remainder  by  means  of  the  capstan -headed  screws 
of  the  level  tubes.  Repeat  the  operation  until  the  bubbles  remain 
in  the  center  of  the  tubes  in  both  positions  of  the  alidade. 

3rd.     To  make  the  line  of  collimation  perpendicular  to  the 
horizontal  axis  of  the  telescope. 


178  PLANE    SURVEYING 

Level  the  table  and  point  the  telescope  towards  some  small 
and  well-defined  object.  Remove  the  screws  which  confine  the 
axis  of  the  telescope  in  its  bearings,  reverse  the  telescope  in  its 
bearings,  that  is,  change  the  axis  end  for  end.  being  careful  not  to 
disturb  the  position  of  the  alidade  upon  the  table,  and  again  sight 
upon  the  same  object.  If  the  intersection  of  the  cross  hairs  bisects 
the  object,  the  adjustment  is  complete.  If  not,  correct  one-half 
of  the  error  by  means  of  the  horizontal  screws  attached  to  the 
reticle.  Sight  on  the  object  again  and  repeat  the  operation  until 
the  line  of  collimation  will  bisect  the  object  in  both  positions  of 
the  telescope. 

4th.  To  make  the  line  of  collimation  parallel  to  the  axis  of 
the  bubble  tube. 

Attach  the  longitudinal  striding  level  to  the  telescope  and 
carry  out  the  adjustment  by  the  "  peg  "  method  as  described  for 
the  transit. 

5th.  To  make  the  horizontal  axis  of  the  telescope  parallel  to 
the  plane  of  the  table. 

Level  the  table  and  point  the  telescope  to  a  well-defined  mark 
at  the  top  of  some  tall  object,  as  near  as  possible  consistent  with 
distinct  vision.  Turn  the  telescope  on  its  horizontal  axis,  and 
point  to  a  small  mark  at  the  base  of  the  same  object.  Draw 
lines  on  the  table  at  the  edge  and  ends  of  the  ruler.  Reverse 
on  these  lines,  point  the  telescope  to  the  lower  object  and  turn  the 
telescope  upon  its  horizontal  axis.  If  the  line  of  collimation  again 
covers  the  higher  point,  the  adjustment  is  complete.  If  it  does 
not,  correct  one-half  of  the  error  by  means  of  the  screws  at  one 
end  of  the  horizontal  axis. 

6th  To  make  the  vertical  arc  or  circle  read  zero  when  the 
line  of  collimation  is  horizontal. 

Level  the  table  and  measure  the  angle  of  elevation  or  depres- 
sion of  some  object.  Remove  the  table  to  the  object,  level  as 
before,  and  measure  the  angle  of  depression  or  elevation  of  the 
first  point  Half  the  difference,  if  any,  of  the  readings  is  the 
error  of  the  adjustment.  Correct  this  by  means  of  the  screws 
attached  to  the  vernier  plate,  and  repeat  the  operation  until  the 
angles  as  read  from  the  two  stations  are  equal. 


186 


PLANE    SURVEYING 


173 


Fig.  125. 


Use.  The  plane-table  is  used  for  the  immediate  mapping  of 
a  survey  made  with  it,  no  angles  being  measured,  but  the  direction 
and  length  of  lines  being  plotted  at  once,  upon  the  paper.  The 
simplest  case  is  the  location  of  a  number  of  points  from  one  central 

point,  called  the  method  of  radi- 
ation. The  table  is  "set  up"  so 
that  some  convenient  point  upon 
the  paper  is  over  a  selected  spot 
upon  the  ground  and  is  then 
clamped  in  azimuth.  Mark  the 
point  upon  the  table  by  sticking  a 
needle  into  the  board.  Now  bring 
the  edge  of  the  alidade  in  contact 
with  the  needle  and  swing  it 
around  until  the  line  of  sight, 

which  is  parallel  to  the  edge  of  the  ruler,  is  directed  to  the  point  to 
be  located.  Having  determined  the  scale  of  the  plat,  aline  is  drawn 
along  the  edge  of  the  ruler  to  scale,  equal  to  the  distance  to  the 
desired  point,  such  distance  having  been  measured  either  with  the 
tape  or  stadia.  In  the  same  way  locate  all  of  the  other  points, 
which  may  include  houses,  trees,  river  banks,  etc.  If  the  plane- 
table  is  set  up  in  the  interior  of  a  field  at  a  point  from  which  all 
of  the  corners  are  visible,  the  corners  can  be  thus  located  and  after 
being  connected,  there  results  a 
plot  of  the  area.  Instead  of  occu- 
pyinga  point  in  the  interior  of  the 
field,  one  corner  may  be  selected 
from  which  all  of  the  others  are 
visible,  or  a  point  outside  of  the 
field  may  be  chosen  from  which  to 
measure  the  lines  to  the  several 
corners.  Evidently  from  such  a 
survey,  data  is  lacking  from  which 
to  calculate  the  area,  and  either 
the  map  must  be  scaled  for  addi- 


Fig.  126. 


tional  data  or  the  area  measured  with  the  planimeter. 

The  Fig.  125  illustrates  the  method  of  surveying  a  closed 
area  by  the  method  of  radiation.     The  plane-table  is  at  the  point 


180 


PLANE    SURVEYING 


o  and  drawn  to  an  exaggerated  scale.  The  area  abode  representing 
to  scale,  the  area  ABODE.  It  may  be  desirable  to  set  up  the 
table  at  some  other  point,  as  for  instance  one  of  the  corners  of  the 
field,  and  run  out  some  of  the  lines  to  the  other  corners  as  a  check 
upon  the  work. 

Traversing,  or  the  Method  of  Progression.  This  method  is 
practically  the  same  as  the  method  of  surveying  a  series  of  lines 
with  the  transit,  but  requires  that  all  of  the  points  be  accessible. 
It  is  the  best  method  of  working  as  it  provides  a  complete  check 
upon  the  survey. 

Let  ABODE,  Fig.  126,  be  the 
series  of  lines  to  be  surveyed  by 
traversing.  Set  up  the  table  at 
B,  the  second  angle  of  the  line, 
so  that  the  point  b  upon  the  paper 
will  be  directly  over  the  point 
B  upon  the  ground.  (The  point 
b  should  be  so  chosen  as  to  leave 
room  upon  the  paper  for  as  much 
of  the  traverse  as  possible.)  Stick 
a  needlo  at  the  point  b  and  place 

the  edge  of  the  alidade  against  it.  Swing  the  alidade  around  until 
the  line  of  sight  covers  the  point  A.  Measure  BA  and  lay  it  off  to 
the  proper  scale  as  ba.  Now  turn  the  alidade  around  the  point  b 
and  sight  to  and  measure  the  distance  BC  and  plot  it  to  scale  as  be. 
Remove  the  instrument  to  c  with  the  point  c  upon  the  paper  directly 
over  C  upon  the  ground,  and  c  b  in  the  direction  of  CB.  This  is  diffi- 
cult to  accomplish  with  the  plane-table,  but  if  the  plot  is  drawn  to 
a  large  scale,  it  must  be  done.  If  the  plot  is  drawn  to  a  small  scale, 
it  will  be  sufficiently  accurate  to  set  the  table  over  the  point  C  as 
nearly  as  possible  in  the  proper  direction  and  then  turn  the  board 
in  azimuth  until  b  is  in  the  direction  of  B.  Stick  a  needle  at  c  and 
check  the  length  of  cb.  Swing  the  alidade  around  c  until  the  line 
of  sight  covers  D,  measure  CD  and  plot  cd.  Remove  to  D  and 
proceed  as  before  and  so  on  through  the  traverse. 

If  the  survey  is  of  a  closed  field,  the  accuracy  of  the  work 
will  be  checked  by  the  closure  of  the  survey. 


Fig.  127. 


188 


PLANE    SURVEYING  181 


The  method  of  progression  is  especially  adapted  to  the  survey 
of  a  road,  the  banks  of  a  river,  etc.,  and  often  many  of  the  details 
may  be  sketched  in  with  the  eye. 

When  the  paper  is  tilled,  put  on  a  new  sheet,  and  on  it,  fix 
two  points,  such  as  D  and  E,  which  were  on  the  former  sheet  and 
from  them  proceed  as  before.  The  sheets  can  afterward  be  united 
so  that  all  points  on  both  shall  be  in  their  true  relative  positions. 

flethod  of  Intersection.  This  is  the  most  rapid  method  of 
using  the  plane-table.  Set  up  the  instrument  at  any  convenient 
point,  as  A  in  Fig.  127  and  sight  to  all  the  desired  points  as  D,  E, 
F,  etc.,  which  are  visible,  and  draw  indefinite  lines  in  their  direc- 
tions. Measure  any  line  as  AB,  B  being  one  of  the  points  sighted 
to,  and  plot  the  length  of  this  line  upon  the  paper  to  any  convenient 
scale.  Move  the  instrument  to  B  so  that  b  upon  the  paper  will  be 
directly  over  B  upon  the  ground,  and  so  that  la  upon  the  paper 
will  be  in  the  direction  of  BA  upon  the  ground  as  explained  under 
the  method  of  progression.  Stick  a  needle  at  the  point  1)  and 
swing  the  alidade  around  it,  sighting  to  all  the  former  points  in 
succession,  and  draw  lines  in  their  direction.  The  intersection  of 
these  two  sets  of  lines  to  the  several  points  will  determine  the 
position  of  the  points.  Connect  the  points  as  d,  e,f,  g,  in  the 
figure.  In  surveying  a  field,  one  side  may  be  taken  as  the  base 
line.  In  choosing  the  base  line,  care  must  be  exercised  to  avoid 
very  acute  or  obtuse  angles:  30°  and  150°  being  the  extreme  limits. 
The  impossibility  of  always  doing  this,  sometimes  renders  this 
method  deficient  in  precision. 

TOPOGRAPHICAL  SURVEYING. 

A  topographical  map  is  one  showing  the  configuration  of  the 
surface  of  the  ground  of  the  area  to  be  mapped  and  includes  lakes, 
rivers,  and  all  other  natural  features,  and  sometimes  artificial 
features  as  well. 

A  topographical  survey  is  one  conducted  for  the  purpose  of 
acquiring  information  necessary  for  the  production  of  a  topograph- 
ical map  of  the  area  surveyed. 

Nearly  all  engineering  enterprises  involve  a  topographical 
survey  more  or  less  extended,  depending  upon  the  nature  and 


182  PLANE   SURVEYING 


importance  of  the  contemplated  work.  The  construction  of  an 
important  building  may  involve  a  survey  of  the  foundation  site 
to  determine  the  amount  of  cut  and  fill ;  the  construction  of  a 
bridge  will  involve  a  hydrographic  survey  of  a  body  of  water  to 
acquire  information  in  regard  to  direction  and  velocity  of  current, 
depth  of  water,  nature  of  bottom,  and  proper  site  for  piers  and 
abutments.  A  proposed  railroad  will  not  only  involve  a  survey 
of  the  line  itself,  but  a  topographical  survey  extending  from  200 
to  400  feet  upon  each  side.  The  design  of  a  sewer  system  or  a 
waterworks  system,  dams,  reservoirs,  canals,  irrigation  channels, 
tunnels,  etc ,  all  involve  topographical  surveys. 

In  what  follows  it  is  intended  to  outline  the  methods  of  con- 
ducting field  operations,  based  partly  upon  the  nature  and  impor- 
tance of  the  problem  involved,  and  partly  upon  the  instruments 
used.  The  different  methods  of  representing  topography  and  the 
involved  drafting-room  work  will  be  fully  treated  in  Topographical 
Drawing. 

The  field  operations,  in  so  far  as  the  methods  and  instruments 
are  concerned,  may  be  classified  as  follows  : 

1.  Sketching  by  the  eye,  without  or  with  the  tape  for  measuring  dis- 
tances. 

2.  Sketching  with  the  aid  of  the  Locke  hand-level  or  clinometer,  hori- 
zontal distances  being  measured  either  by  pacing  or  with  the  tape. 

3.  Determining  the  elevation  of  points  with  the  wye-level,  horizontal 
distances  being  determined  either  with  the  stadia  or  tape. 

4.  Determining  points  with  the  transit  and  stadia. 

5.  Topographical  sketching  with  the  plane-table  and  stadia. 

6.  Photography. 

7.  Triangulation. 

It  is  evident  that  the  first  method  is  entirely  lacking  in  accu- 
racy, and  such  work  should  be  done  only  when  speed  is  the  most 
important  consideration,  only  the  roughest  approximation  to  the 
topographical  features  being  attempted  ;  contour  lines  cannot  be 
located.  Work  of  this  nature  is  of  value  principally  for  purposes 
of  promoting  an  enterprise  ;  artistic,  showy  plates  being  desired. 
Little  can  be  said  descriptive  of  the  manner  of  carrying  out  the 
field  work,  since  this  will  require  considerable  artistic  ability  as 
well  as  the  ability  to  "  see  "  things  and  estimate  distances.  Com- 
paratively  few  men  possess  the  ability  to  carry  out  topography  of 
this  nature.  It  necessarily  follows  that  the  work  must  be  done 


PLANE    SURVEYING  183 

entirely   by   sketching  in    the   field,    and    for   this    purpose    the 
following  equipment  is  needed  : 

2  or  3  medium  pencils,  kept  well  sharpened. 

Rubber  eraser. 

Thumb-tacks. 

Several  sheets  of  drawing  paper,  14"  x  14". 

One  light  drawing  board,  15"  x  15". 

A  pocket  compass  will  be  useful  in  determining  the  bearing 
to  prominent  objects  to  tie  in  the  stations  of  the  survey.  A  Locke 
hand-level  or  Abney  clinometer  will  also  be  useful  for  finding 
approximate  heights,  and  either  of  these  instruments  can  be 
readily  carried  in  the  pocket.  It  will  be  more  convenient  to  have 
the  paper  cross-ruled  into  one-fourth  inch  squares,  the  center  line 
being  ruled  in  red,  but  if  drawing  paper  is  used,  it  will  be  neces- 
sary to  add  an  engineer's  scale  to  the  equipment.  The  back  of 
the  drawing  board  should  be  fitted  with  a  leather  pocket,  with  flap 
and  button,  in  which  the  blank  sheets  and  the  finished  topographic 
sheets  should  be  kept.  A  strap  attached  to  the  board  and  to  go 
over  the  shoulder,  will  prove  a  great  convenience.  A  waterproof 
cover  should  be  provided  to  protect  the  board  and  sheets  in  case 
of  rain. 

A  compass  or  transit  survey  forms  the  backbone  of  the  topog- 
raphy, and  the  sketching  should  include  an  area  upon  each  side  of 
the  line  so  surveyed,  and  running  parallel  with  it. 

A  separate  sheet  should  be  used  for  each  course  (by  course  is 
intended  the  straight  line  from  one  turning  point  to  the  next),  no 
matter  how  short  it  may  be.  Begin  at  the  bottom  of  the  sheet 
and  sketch  the  topography  up  the  sheet,  that  is,  in  the  direction 
of  the  progress  of  the  survey,  and  number  the  sheets  in  order. 
Begin  each  new  sheet  with  the  same  station  that  ended  the  preced- 
ing sheet.  After  the  field  work  is  completed,  the  sheets  can  be 
laid  down  in  order,  the  angles  between  their  center  lines  corre- 
sponding to  the  deflection  angles  as  given  by  the  transit  notes  of 
the  survey.  The  topography  can  now  be  traced  upon  tracing  cloth 
in  a  continuous  sheet.  The  method  above  outlined  will  result  in 
a  saving  of  time,  especially  in  wrorking  up  the  topographic  plat. 

The  second  method  commends  itself  in  connection  with  a 
preliminary  survey  of  a  highway,  steam  or  electric  road,  irriga- 


184  PLANE   SURVEYING 

tion  channels,  canals,  -etc.     The  equipment  should  be  as  follows  : 

1  or  2  straight  edges,  about  12  feet  in  length. 
1  or  2,  100-foot  steel  tapes. 
1  or  2  plumb-bobs. 

1  pocket  compass. 

2  or  3  medium  pencils,  kept  well  sharpened. 
Rubber  eraser. 

Thumb-tacks. 

Several  sheets  of  drawing  paper  or  cross-section  paper,  14"  X  14". 

One  light  drawing  board,  15"  X  15"  with  waterproof  cover. 

The  topographic  party  should  be  made  up  of  the  topographer 
arid  one  or  two  assistants,  depending  somewhat  upon  the  nature  of 
the  survey  and  the  country  traversed.  If  the  country  permits  of 
rapid  progress  of  the  transit  and  level  party,  two  assistants  will  be 
necessary  to  keep  the  topography  abreast  of  the  survey.  Rapid  work 
may,  however,  be  done  with  one  assistant,  provided  the  topography 
does  not  extend  more  than  200  feet  each  side  of  the  transit  line. 

The  Abney  clinometer  is  well  adapted  for  this  class  of  work, 
on  account  of  its  portability,  which  is  an  important  item  in  a 
rough  country  with  steep  side  slopes.  It  can  be  used  in  the  same 
way  as  the  Locke  hand-level,  if  necessary,  but  is  a  more  generally 
useful  instrument,  as  is  described  in  Part  1.  The  straight  edge 
should  be  of  well-seasoned,  straight-grained  material,  as  light  as 
possible,  but  so  constructed  as  to  prevent  warping.  It  should  be 
divided  into  spaces  of  one  foot  each,  painted  alternately  red  and 
white.  The  tapes  should  be  of  band  steel,  as  they  are  subjected 
to  rough  usage,  and  they  should  be  divided  to  feet  and  tenths  at 
least.  A  plumb-bob  is  necessary  for  plumbing  down  the  end  of 
the  tape  on  steep  slopes.  The  pocket  compass  is  a  necessary 
adjunct  in  work  of  this  character.  The  drawing  paper  should 
preferably  be  cross-section  paper  ruled  into  one-fourth  inch  squares 
with  a  heavy  center  line  in  red,  but  if  ordinary  drawing  paper  is 
used,  it  will  be  necessary  to  include  in  the  outfit  an  engineer's 
scale,  by  means  of  which  distances  may  be  platted  upon  the  sheet. 
Enough  of  these  sheets  should  be  carried  to  cover  a  day's  work,  but 
no  more.  The  drawing  board  should  be  fitted  up  as  described 
tinder  the  previous  method. 

Method  of  Procedure.  The  transit  line  furnishes,  of  course, 
the  backbone  of  the  survey,  and  the  topography  will  be  taken  for 


192 


PLANE    SURVEYING  185 


the  proper  distance  upon  each  side  of  this  line,  by  locating  points 
both  as  to  distance  and  elevation,  upon  perpendiculars  from  the 
transit  stations.  In  rough  country,  it  may  be  necessary  to  locate 
these  points  intermediate  between  the  transit  stations.  Before 
starting  out  upon  a  day's  work  it  is  necessary  to  procure  from  the 
level  party,  the  elevation  of  the  transit  stations,  or  if  the  topog- 
raphy keeps  pace  \dth  the  transit  survey,  the  elevation  may  be  ob- 
tained from  the  leveler  at  each  station.  For  points  intermediate 
between  transit  stations,  the  elevations  may  be  gotten  closely 
enough  with  the  clinometer  or  hand-level.  The  number  of  each 
station  as  well  as  its  elevation,  should  be  noted  upon  the  topo- 
graphic sheet,  and  the  topography  will  include  the  location  of 
contour  lines,  at  proper  vertical  intervals,  as  well  as  all  streams, 
lakes,  property  lines,  etc.  An  example  showing  the  method  of 
keeping  the  field  notes,  will  at  the  same  time  best  serve  to  explain 
the  methods  of  conducting  the  survey. 

Beginning  with  station  0  at  the  bottom  of  the  sheet,  the 
number  and  elevation  of  the  station  are  noted.  See  Fig.  128. 
Sending  the  assistant  out  upon  one  side  of  the  transit  line  and  at 
right  angles  thereto,  he  holds  the  rod  at  points  to  be  designated  by 
the  topographer,  the  distances  to  be  determined  by  pacing,  or  with 
the  tape,  and  the  elevations  determined  either  by  sighting  upon 
the  rod  with  the  clinometer,  or  by  laying  the  straight  edge  upon 
the  ground  at  right  angles  to  the  line  and  applying  the  clinometer 
to  it  to  determine  the  slope,  from  which  elevations  can  at  once  be 
determined.  Contour  points  are  then  readily  interpolated  and  the 
distance  out  platted  to  scale  upon  the  sheet  and  a  note  made  of  the 
elevation  of  the  contour  lines.  If  a  lake  or  stream  intervenes 
within  the  limits  of  the  topographic  survey,  determine  the  distance 
to  and  elevation  of  the  shore  line  and  plat  upon  the  sheet.  Deter- 
mine points  upon  the  other  side  of  the  transit  line  in  the  same  way. 

If  one  or  more  contour  lines  cross  the  transit  line  between 
stations,  determine  the  points  of  crossing  and  plat  the  points  upon 
the  sheet,  to  scale,  as  shown  between  stations  0  and  1.  It  will  be 
noticed  in  this  case  that  the  elevation  of  station  0,  is  138  feet  and 
of  station  1,  is  141  feet.  If  contours  are  to  be  taken  at  vertical 
intervals  of  five  feet,  it  is  apparent  that  the  140-foot  contour  line 
must  cross  the  transit  line  between  these  stations.  If  the  slope  of 


186                                   PLANE    SURVEYING 
A       


B  C 

Fig.  128. 

the  ground  is  uniform,  the  point  of  crossing  may  be  taken  at  two- 
thirds  of  the  distance  from  0  to  1.  Otherwise,  locate  the  point 
with  the  clinometer. 


PLANE    SURVEYING 


187 


\ 


\ 


\ 


145 


\ 


140 


\ 


Fig.  128. 

Now  go  to  station   1  and  locate  contour  points  and  other 
topographic  features  as  before  described,  and  connect  points  in  the 


188  PLANE    SURVEYING 

same  contour  line,  sketching  in  the  curve  of  the  line  with  the  eye. 
Use  a  separate  sheet  for  each  portion  of  the  transit  line  from  turn- 
ing point  to  turning  point;  this  will  require  that  the  turning 
points  appear  upon  two  consecutive  sheets.  Likewise,  if  the  length 
of  the  line  between  turning  points  is  too  long  to  be  platted  upon  a 
single  sheet,  begin  the  second  sheet  with  the  same  station  that 
completed  the  first  sheet  and  so  continue  throughout  the  survey. 
As  each  sheet  is  completed,  number  it  and  return  to  the  pocket  on 
the  back  of  the  drawing  board.  The  pocket  compass  should  be 
used  for  determining  the  bearing  of  property  lines,  roads,  streams, 
etc.,  crossed  by  the  survey,  and  to  take  the  bearings  to  prominent 
objects. 

The  topographic  sheets  should  be  filed  away  in  such  a  manner 
as  to  make  them  easily  accessible  at  any  time,  as  the  engineer  in 
charge  of  the  transit  survey  may  wish  to  consult  them  from  time 
to  time.  The  office  work  of  preparing  the  topographic  plat  can  be 
very  expeditiously  carried  out  as  before  described. 

The  use  of  the  wye-level  as  a  topographic  instrument  is 
limited,  but  for  certain  kinds  of  work  the  instrument  is  the  most 
satisfactory,  as  for  instance,  the  survey  of  a  dam-site;  the  survey 
of  a  reservoir-site;  the  survey  of  a  town  preparatory  to  planning 
sewer  and  waterworks  systems  and  the  planning  of  street  pave- 
ments. 

The  instrument  should  be  fitted  with  stadia  wires  for  measur- 
ing horizontal  distances,  and  this  will  usually  prove  a  great  conven- 
ience, resulting  in  saving  of  both  time  and  expense.  A  steel 
tape  should,  however,  be  included  in  the  equipment  for  field  work, 
for  the  purpose  of  checking  measurements  with  the  stadia.  In 
addition  to  the  above  there  should  be  provided,  the  following 
equipment: 

Self-reading  level  rod,  capable  of  being  read  to  hundredths  of  a  foot. 

Hatchet. 

Marking  crayon. 

2  or  3  medium  pencils,  kept  well  sharpened. 

Plumb-bob. 

Rubber  eraser. 

Portable  turning  point. 

The  method  of  using  the  level  rod  in  connection  with  the 
stadia  for  measuring  distances  has  been  fully  discussed  in  Part  II. 


PLANE    SURVEYING  189 

The  portable  turning  point  will  prove  of  great  convenience 
and  may  be  made  from  a  triangular  piece  of  thin  steel,  with  the 
corners  turned  down  to  project  about  one  inch. 

If  the  level  is  to  be  used  with  the  tape,  the  party  will  be 
made  up  of  the  levelman,  two  tapemen  and  a  rodman,  unless  the 
nature  of  the  work  will  permit  of  the  rodman  carrying  the  rear 
end  of  the  tape.  If  the  stadia  is  used  for  measuring  distances, 
only  the  rodman  will  be  required  in  addition  to  the  levelman. 
The  levelman  carries  the  note  book  and  enters  into  it  all  rod 
readings  both  for  elevation  and  distances.  These  notes  should  be 
entered  upon  the  left-hand  page,  the  right-hand  page  being  re- 
served for  notes  and  sketches,  which  should  be  as  full  as  possible. 
The  levelman  should  cultivate  the  practice  of  calculating  the  ele- 
vations of  the  stations  as  the  work  progresses,  at  least  of  the 
turning  points  and  bench-marks,  in  order  that  the  results  may  be 
checked  and  errors  discovered  at  once  and  corrected.  If  this  work 
is  left  to  be  afterward  carried  out  in  the  office,  errors  may  be  dis- 
covered that  may  require  considerable  time  to  locate  and  correct. 

If  the  area  to  be  surveyed  is,  for  instance,  a  reservoir  site,  it 
will  be  found  most  convenient  to  cover  the  area  with  a  system  of 
rectangles  as  shown  in  the  figure,  the  parallel  lines  being  spaced 
from  200  to  400  feet  apart  as  may  be  most  desirable.  These  lines 
should  be  run  in  with  the  transit,  stakes  being  set  at  the  inter- 
sections of  the  cross  lines,  or  if  the  area  is  not  very  extended  and 
is  comparatively  level,,  by  means  of  the  level  itself,  the  perpen- 
dicular distances  between  the  parallel  lines  being  measured  with 
the  tape. 

These  lines  having  been  laid  down,  the  next  step  is  to  estab- 
lish a  system  of  bench-marks  over  the  area.  Begin  by  establishing 
a  "standard"  bench-mark  at  some  central  point  upon  a  permanent 
object,  easily  identified,  and  from  thence  radiate  in  all  directions, 
returning  finally  to  the  original  bench-mark  for  purposes  of 
checking.  Having  located  and  satisfactorily  checked  the  bench- 
marks, begin  by  running  the  level  over  all  the  lines  running  in 
one  direction,  as  from  A  to  B,  back  from  C  to  D  and  so  on,  taking 
rod  readings  at  every  fifty  or  one  hundred  feet,  in  addition  to  the 
readings  at  the  stakes  at  intersections  of  cross  lines.  It  is  to  be 
understood  that  stakes  are  not  to  be  driven  at  the  intermediate 


190  PLANE    SURVEYING 

points.  Next  run  the  level  over  the  lines  at  right  angles  to  the 
former  ones  and  in  the  same  way,  checking  the  levels  at  inter- 
sections. Advantage  should  be  taken  of  every  opportunity  to 
check  upon  bench-marks  previously  located,  and  to  establish 
others. 

In  keeping  the  field  records,  the  notes  of  the  two  sets  of  lines 
should  be  kept  in  separate  books  ;  that  is  to  say,  if,  for  instance, 
one  set  of  lines  run  north  and  south,  and,  therefore,  the  other  east 
and  west,  the  notes  of  the  north  and  south  lines  should  be  entered 
in  one  set  of  books  and  the  notes  of  the  east  and  west  lines  in  an- 
other set,  and  a  note  should  be  made  of  the  direction  in  which  a 
line  is  run,  as  from  north  to  south  or  from  east  to  west. 

In  conducting  a  survey  for  the  preparation  of  a  topographical 
map  necessary  to  the  design  of  a  sewer  or  waterworks  system, 
much  the  same  method  is  to  be  followed,  but  now  the  streets  and 
alleys  take  the  place  of  the  rectangular  system  referred  to  above. 
As  before,  all  the  streets  and  alleys  running  in  parallel  directions 
are  to  be  gone  over  in  a  systematic  way,  readings  being  taken  fifty 
or  one  hundred  feet  apart  in  addition  to  street  and  alley  intersec- 
tions. (By  street  and  alley  intersections  is  intended  the  intersec- 
tions of  the  center  lines,  the  lines  of  levels  being  run  along  these 
center  lines.)  If  a  fairly  accurate  map  of  a  town  is  available,  the 
distances  measured  with  the  tape  along  the  center  line  of  the  streets 
and  alleys  will  serve  as  a  check  upon  the  map.  If,  however,  dis- 
crepancies occur  or  there  is  no  map  available,  it  will  be  necessary 
to  use  the  transit  for  staking  out  street  lines  and  for  determin- 
ing the  relative  directions  of  these  lines.  It  follows  that  the 
topography  of  the  ground  between  streets  and  alleys  can  only  be 
approximated,  but  sufiicient  points  accurately  determined  will 
have  been  established  to  permit  the  platting  of  a  contour  map, 
from  which  the  system  can  be  laid  down. 

The  office  work  involved  in  the  survey  of  an  area,  as  above 
described^  consists  in  preparing  profiles  of  the  level  lines  and  pre- 
paring a  plat  of  the  lines  surveyed.  From  the  profiles  the  contour 
points  can  be  laid  down  in  their  proper  position  upon  the  plat,  and  as 
each  point  is  laid  down,  its  elevation  should  be  noted  in  pencil,  and 
after  all  the  points  have  been  platted,  the  points  in  the  same  contour 
line  can  be  connected — preferably  free-hand — producing  the  con- 


PLANE    SURVEYING  191 

tour  map.  The  scale  to  be  adopted  will  depend  upon  the  nature 
of  the  work,  but  should  be  as  large  as  possible,  consistent  with  the 
convenient  handling  of  the  map. 

Transit  and  Stadia.  The  method  by  transit  and  stadia  is  of 
more  general  application  than  the  preceding  method,  points  being 
located  by  "  polar  co-ordinates,"  that  is  to  say,  by  direction  and 
distance  from  a  known  point,  the  elevation  being  determined  at 
the  same  time. 

Method  of  conducting  field  operations.  If  the  area  to  be  sur- 
veyed is  small,  the  preceding  method,  based  upon  a  system  of 
rectangles,  will  prove  satisfactory,  and  the  elevations  of  the  corners 
and  salient  points  can  be  determined  at  the  same  time  that  the 
lines  forming  the  rectangles  are  being  laid  down.  Especial  care 
should  be  taken  to  check  the  elevations  of  the  corners. 

In  making  a  survey  for  a  sewer  or  a  waterworks  system,  the 
transit  and  stadia  method  will  be  found  efficient,  especially  in  cases 
where  no  survey  has  previously  been  made,  the  map,  if  it  exists  at 
all,  having  been  compiled  from  the  records  in  the  County  Record- 
er's office.  The  bench-marks  necessary  in  a  survey  of  this  kind, 
however,  should  be  established  with  the  wye-level,  and  it  may  be 
desirable  to  determine  the  elevation  of  street  intersections  in  the 
same  way. 

If  the  area  to  be  surveyed  is  too  large,  or  of  uneven  topography, 
proceed  as  follows:  Choose  a  point,  as  the  intersection  of  two 
streets,  the  corner  of  a  farm,  or  an  arbitrary  point  conveniently 
located  and  drive  a  stake  firmly  at  this  point,  "witnessing" 
it  from  other  easily  recognized  points  or  stakes.  The  transit  should 
be  set  over  this  point  with  the  vernier  reading  zero,  and  the  instru- 
ment pointed  by  the  lower  motion  in  the  direction  of  the  meridian. 
This  may  be  the  true  meridian  previously  determined,  the  mag- 
netic meridian  as  shown  by  the  needle,  or  an  arbitrary  meridian 
assumed  for  the  purpose  of  the  survey.  It  will  generally  be  more 
satisfactory  to  run  out  a  true  meridian  by  means  of  the  solar 
attachment,  but  in  any  event  the  direction  of  the  line  taken  as  a 
meridian  should  be  defined  by  stakes,  firmly  driven  into  the  ground, 
and  "witnessed"  by  stakes  or  other  objects  easily  recognized. 
The  elevation  of  .the  starting  point,  if  not  known,  is  assumed  and 
recorded  in  the  notebook.  A  traverse  line  should  now  be  run,  its 


192  PLANE    SURVEYING 


position  and  direction  chosen  with  a  view  to  obtaining  from  each 
station  the  largest  possible  number  of  pointings  to  salient  features 
of  the  area  under  survey,  and  these  pointings  are  taken  while  the 
instrument  is  set  at  any  station,  and  before  the  traverse  is  com- 
pleted. 

The  length  of  each  course  is  measured  with  the  stadia,  and 
together  with  the  azimuth  and  the  vertical  angle,  it  should  be 
recorded  in  the  notebook.  The  length,  azimuth,  and  vertical  angle 
of  each  course  should  be  read'  from  both  ends  to  serve  as  a  check. 
The  additional  pointings  taken  from  each  course  of  a  traverse  are 
usually  called  "  side  shots",  and  for  each  there  are  required  the 
distance,  azimuth,  and  vertical  angle.  These  will  locate  the  point 
and  determine  its  elevation. 

The  method  of  using  the  stadia  has  already  been  quite  fully 
discussed  in  Part  II.,  and  need  not  be  repeated. 

The  points  selected  for  side  shots  should  be  such  as  will 
enable  the  contours  to  be  platted  intelligently  aLd  accurately  upon 
the  map  of  the  area  under  survey.  They  should  be  taken  along 
ridges  and  hollows  and  at  all  changes  of  slope.  They  should  be 
taken  at  frequent  intervals  along  a  stream  to  indicate  its  course, 
or  along  the  shore  of  a  lake.  It  is  usually  required  that  the 
location  of  artificial  structures,  such  as  houses,  fences,  roads,  etc., 
be  determined  that  they  may  be  mapped  in  their  proper  position. 
Pointings,  therefore,  should  be  taken  to  all  fence  corners  and 
angles,  and  to  enough  corners  and  angles  of  buildings,  to  permit 
of  their  being  platted.  Sufficient  points  should  be  taken  along 
roads  to  determine  their  direction.  Wooded  lands,  swamps,  etc., 
may  be  indicated  by  pointings  taken  around  their  edges.  In 
addition  to  the  notes  above  described,  the  recorder  should  amplify 
the  notes  with  sketches,  to  aid  the  memory  in  mapping. 

The  traverse,  of  course,  forms  the  backbone  of  such  a  survey, 
and  the  accuracy  of  the  resulting  topographical  map  will  depend 
upon  the  degree  of  care  bestowed  upon  running  the  courses. 
Over  uneven  ground,  it  is  often  desirable  to  run  a  secondary  trav- 
erse from  the  first,  for  the  more  rapid  and  accurate  location  of 
points. 

The  organization  of  a  party  will  depend  upon  the  nature 
of  the  country  traversed  and  of  the  results  required.  Changes  in 


PLANE    SURVEYING  193 

the  make-up  of  parties,  as  given  below,  will  suggest  themselves 
for  any  special  work. 

For  economy  and  speed,  the  party  for  taking  topography  with 
transit  and  stadia  will  consist  of  a  transitman  or  observer,  a 
recorder  in  charge  of  the  notebook,  who  should  be  capable  of  making 
such  sketches  as  are  necessary,  and  two  to  four  men  with  stadia 
rods.  The  greater  the  distances  to  be  traversed  by  the  stadia  men 
between  points  taken,  the  greater  number  the  observer  can  work 
to  advantage.  One  or  two  axemen  may  be  employed  if  clearing 
is  to  be  done. 

The  party  may  be  reduced  to  two  men — one  to  handle  the 
instrument,  record  notes  and  make  sketches,  the  other  to  carry  the 
rod. 

The  Plane  Table  and  Stadia.  The  plane  table  is  an  instru- 
ment intended  for  topographic  purposes  only  and  is  used  for  the 
immediate  mapping  of  a  survey  made  with  it,  no  notes  of  angles 
being  taken,  but  the  lines  being  platted  at  once  upon  the  paper. 
The  use  of  the  plane  table  has  been  fully  described.  In  topo- 
graphical work  over  an  extended  area,  it  may  be  used  for  filling  in 
details,  based  upon  a  previous  traverse  made  with  a  transit,  or 
based  upon  a  system  of  triangulation  as  will  be  described.  Over 
small  areas,  the  traverse  itself  may  be  run  with  the  plane  table 
and  the  details  filled  in  at  the  same  time.  It  is  the  standard  in- 
strument of  the  United  States  Geological  Survey  and  is  largely 
used  upon  the  United  States  Geodetic  Survey. 

The  points  in  favor  of  the  plane  table  are  :  Economy,  since 
the  map  is  made  at  once  without  the  expense  of  notes  and  sketches; 
and  as  the  mapping  is  all  done  upon  the  ground  to  be  represented, 
all  of  its  peculiarities  and  characteristics  can  be  correctly  repre- 
sented. 

On  the  other  hand,  the  plane  table  is  an  instrument  useful 
only  for  taking  topography  ;  the  rodmen  are  idle  while  the  map- 
ping is  being  done  ;  the  instrument  is  more  unwieldy  than  the 
transit,  particularly  upon  difficult  ground  ;  the  record  of  the  work 
for  a  long  period  is  constantly  exposed  to  accident  ;  the  distortion 
of  the  paper  with  the  varying  dampness  of  air,  introduces  errors 
in  the  map  ;  while  the  area  exposed  makes  it  too  unstable  to  use 
in  high  winds. 


194  PLANE    SUKVEYING 

The  organization  of  a  party  for  the  taking  of  topography, 
using  the  plane  table,  is  much  the  same  as  with  the  transit  and 
stadia  ;  however,  on  account  of  the  weight  of  the  instrument, 
means  of  transportation  must  be  provided. 

A  less  number  of  rodmen  can  be  employed  than  with  the 
stadia,  owing  to  the  time  required  for  mapping. 

An  observer,  a  man  to  reduce  stadia  notes  and  sketch  topog- 
raphy around  points  determined  by  intersection  or  stadia  from 
the  plane  table  station,  and  one  rodman,  will  make  the  minimum 
working  party,  in  addition  to  which,  axemen  and  a  team  for  trans- 
portation will  be  required. 

Photography.  The  following  is  taken  from  Gillespies  Sur- 
veying (Staley). 

"  Photography  has  long  been  successfully  employed  by 
European  engineers,  notably  those  of  Italy,  for  the  purpose  of 
taking  topography.  The  Canadian  Government  has  also  employed 
it  successfully  in  the  survey  of  Alaska. 

The  recommendation  of  this  method  is  the  great  saving  of 
time  in  the  field,  while  giving  topographic  features  with  all  the 
accuracy  required  for  maps  to  be  platted  on  a  scale  of  1  to  25,000. 

JV1.  Javary  states  that  the  maximum  error  both  for  horizontal 
distances  and  elevations,  using  a  camera  with  a  focal  length  of 
twenty  inches  and  a  microscope  in  examining  the  points,  was 
only  1  in  5,000  as  deduced  from  a  number  of  cases. 

M.  Laussedat,  in  his  work,  found  that  this  method  did  not 
require  more  than  one-third  the  time  necessary  by  the  usual 
methods. 

This  makes  it  especially  suitable  in  all  mountainous  regions, 
where  so  much  time  is  lost  in  getting  to  and  from  stations,  that 
little  is  available  for  observations  and  sketching. 

A  single  occupation  of  a  station  with  photographic  apparatus 
would  suffice  to  complete  work  that  with  the  ordinary  methods 
would  require  several  days." 

Instruments.  The  ordinary  camera  may  be  used,  if  it  is  pro- 
vided with  a  level.  A  tripod  head  for  leveling  the  instrument, 
and  a  roughly  graduated  horizontal  circle  for  reading  the  direction 
of  the  line  of  sight,  when  photographing  different  parts  of  the 
horizon,  are  convenient  attachments. 


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II 


PLANE   SURVEYING  195 


A  camera  is  sometimes  used  upon  a  plane  table,  the  record 
of  the  work  being*  made  upon  the  paper  in  connection  with  a  set 
of  radial  lines  drawn  from  the  point  representing  the  station 
occupied. 

Many  special  forms  of  instrument  combining  the  camera  and 
theodolite  have  been  devised,  some  one  of  which  should  be  used 
if  work  of  this  kind  is  to  be  undertaken  on  a  large  scale.  For  a 
description  of  these  instruments,  and  a  complete  treatise  on  this 
subject,  comprising  a  discussion  of  the  requirements  of  the  appa- 
ratus, the  fundamental  principles  of  photography,  methods  of  field 
work,  forms  of  notes,  reduction  of  notes  and  making  of  the  map, 
together  with  the  bibliography  of  the  subject,  see  United  States 
Coast  and  Geodetic  Survey  Report,  1893,  Part  II.,  Appendix  3. 

The  camera  tripod  as  ordinarily  constructed  is  too  unstable 
for  purposes  of  topographic  surveying,  and  it  is  desirable  to  have 
a  tripod  constructed  especially  for  this  class  of  work.  Glass  plates 
are  heavy  and  awkward  to  carry  aside  from  their  fragile  nature. 
Cut  films  can  be  procured  in  any  of  the  standard  sizes,  and  as 
they  are  light  and  stand  rough  handling  and  give  ordinarily  as 
good  results  as  the  glass  plates,  they  are  to  be  preferred.  Their 
cost  is  about  double  that  of  the  glass. 

TRIANQULATION. 

This  method  of  surveying  is  sometimes  called  "  Trigonometric 
Surveying"  and  sometimes  "Geodetic  Surveying",  though  this 
latter  is  properly  applied  only  when  the  area  to  be  surveyed  is  so 
extensive  that  allowance  must  be  made  for  the  curvature  of  the 
earth.  Since  this  instruction  paper  is  devoted  to  Plane  Surveying 
only,  the  curvature  of  the  earth  will  be  neglected. 

Triangulation,  or  Triangular  Surveying,  is  founded  upon  the 
method  of  determining  the  position  of  a  point  at  the  apex  of  a- 
triangle  of  which  the  base  and  two  angles  are  measured.  Thus  in 
Fig.  129  the  length  of  the  base  line  AB  is  measured  and  the 
angles  PAB  and  PB  A  are  measured,  from  which  can  be  calculated 
the  lengths  of  the  sides  PA  and  PB.  This  calculated  length  of 
PA  will  then  be  taken  as  the  side  of  a  second  triangle,  and  the 
angles  PAC  and  PCA  measured,  from  which  the  other  sides  of 
the  triangle  can  be  calculated.  By  an  extension  of  this  principle, 


196  PLANE    SURVEYING 


a  field,  farm,  or  a  country  can  be  surveyed  by  measuring  a  base 
line  only,  and  calculating  all  of  the  other  desired  distances,  which 
are  made  the  sides  of  a  connected  series  of  imaginary  triangles 
whose  angles  are  carefully  measured. 

Measuring  the  base  line.  For 
a  base  line,  a  fairly  level  stretch 
of  ground  is  selected,  as  nearly  as 
possible  in  the  middle  of  the  area 
to  be  surveyed,  and  a  line  from 
one  thousand  feet  to  one-half  mile, 
or  longer,  is  very  carefully  meas- 
ured. The  ends  of  this  line  are 
marked  with  stone  monuments  or 
solid  stakes.  If  the  survey  'is  of 
sufficient  importance,  the  ends  of 

the  base  line  and  the  apexes  of  the 
Fig.  129. 

triangles  should  be  permanently 

preserved  by  means  of  stones  not  less  than  six  inches  square  in 
cross-section  and  two  feet  long,  these  stones  being  set  deep  enough 
to  be  beyond  the  disturbing  action  of  frost.  Into  the  top  of  this 
stone  should  be  leaded  a  copper  bolt  about  one-half  inch  in  diam- 
eter, the  head  of  the  bolt  being  marked  with  a  cross  to  designate 
the  exact  point.  The  point  may  be  brought  to  the  surface  by  a 
plumb-line  for  use  in  the  survey.  The  location  of  each  monument 
should  be  fully  described  with  reference  to  surrounding  objects  of 
a  permanent  character,  so  as  to  be  easily  recovered  for  future  use. 

The  measurement  of  the  base  line  for  the  areas  of  limited 
extent  should  be  made  with  a  precision  of  from  one  in  five  thousand 
to  one  in  fifty  thousand,  depending  upon  the  scale  of  the  map, 
the  extent  of  the  area  under  survey,  and  the  nature  and  importance 
of  the  work. 

The  two  ends  of  the  base  line  having  been  determined  and 
marked,  the  transit  is  set  over  one  end  and  a  line  of  stakes  ranged 
out  between  the  two  ends,  especial  care  being  taken  to  make  the 
alignment  as  perfect  as  possible.  These  stakes  should  be  not  less 
than  two  inches  square,  driven  firmly  into  the  ground,  preferably 
at  even  tape  lengths  apart,  or  at  least  at  one-half  or  one-quarter 
tape  lengths,  center  to  center;  the  centers  should  be  marked  by 


204 


PLANE    SURVEYING  19V 

fine  scratches  upon  strips  of  tin  or  zinc  tacked  to  the  top  of  the 
stakes, 

For  ordinary  work  the  base  line  may  be  measured  with  a  tape, 
notes  being  made  of  the  temperature,  pull,  grade,  and  distances 
between  supports,  the  tape  having  been  previously  standardized. 
For  a  degree  of  precision,  such  as  is  attempted  upon  the  work  of 
the  United  States  Coast  and  Geodetic  Survey,  more  refined  methods 
are  used,  but  as  this  properly  belongs  to  geodetic  surveying,  it  is 
unnecessary  to  consider  it  here. 

Measuring  the  angles.  After  establishing  and  measuring 
the  base  line,  prominent  points  are  chosen  for  triangulation  points 
or  apexes  of  triangles,  and  from  the  extremities  of  the  base  line 
angles  are  observed  to  these  points,  care  being  taken  to  BO  choose 
the  points  that  the  angles  shall  in  no  case  be  less  than  30°,  nor 
more  than  120°.  The  distances  to  these  and  between  these  points 
are  then  calculated  by  trigonometric  methods,  the  instrument  being 
then  placed  at  each  of  these  new  stations  and  angles  observed  from 
them  to  still  more  distant  stations,  the  calculated  lines  being  used 
as  new  base  lines.  This  process  is  repeated  and  extended  until  the 
entire  district  included  in  the  survey  is  covered  with  a  network  of 
"primary  triangles  "  of  as  large  sides  as  possible.  One  side  of  the 
last  triangle  should  be  so  located  that  its  length  can  be  determined 
by  direct  measurement  as  well  as  by  calculation ;  the  accuracy  of 
the  work  can  thus  be  checked.  Within  these  primary  triangles 
secondary  or  smaller  triangles'  are  formed  to  serve  as  the  starting 
points  for  ordinary  surveys  with  the  transit  and  tape,  transit  and 
stadia,  plane  table,  etc.,  to  fix  the  location  of  minor  details. 
Tertiary  triangles  may  also  be  formed. 

When  the  survey  is  not  very  extensive,  and  extreme  accuracy 
is  not  required,  the  ordinary  methods  of  measuring  angles  may  be 
employed.  Otherwise  there  are  two  methods  of  measuring  angles, 
called,  respectively,  the  method  of  repetition  and  the  method  by 
continuous  reading.  When  an  engineer's  transit  is  used  for 
measuring  angles,  the  method  by  repetition  is  the  simplest  and 
best  and  is  carried  out  as  follows  :  The  vernier  is  preferably  set 
at  zero  degrees  and  then  by  the  lower  motion  turned  upon  the  left- 
hand  station;  the  lower  motion  is  then  clamped  and  the  instrument 
turned  by  the  upper  motion  upon  the  right-hand  station:  the 


205 


198  PLANE   SURVEYING 

upper  motion  is  then  clamped  and  the  instrument  turned  by  the 
lower  motion  upon  the  left-hand  station;  lower  motion  clamped 
and  instrument  again  turned  by  upper  motion  upon  right-hand 
station.  This  process  is  repeated  as  often  as  may  be  necessary  to 
practically  cover  the  entire  circle  of  360°  and  the  circle  is  then 
read.  This  reading  divided  by  the  number  of  repetitions  will  give 
the  value  of  the  angle. 

Now  reverse  the  telescope  and  repeat  the  observations 
described  above,  but  from  right  to  left;  the  readings  being  taken 
in  both  directions  to  eliminate  errors  due  to  clamping  and  unclamp- 
ing  and  personal  errors  due  to  mistakes  in  setting  upon  a  station. 
The  readings  should  be  taken  with  the  telescope  both  direct  and 
reverse  to  eliminate  errors  of  adjustments.  Both  verniers  should 
be  read  in  order  to  eliminate  errors  due  to  eccentricity  of  verniers, 
and  the  entire  circle  is  included  in  the  operation  in  order  to  elim- 
inate errors  due  to  graduation. 

The  second  method,  by  continuous  reading,  consists  in  point- 
ing the  telescope  at  each  of  the  stations  consecutively,  and  reading 
the  vernier  at  each  pointing;  the  difference  between  the  consecu- 
tive readings  being  the  angle  be- 
tween  the  corresponding  points. 
Thus  in  Fig.  130  with  the  instru- 
ment at  zero,  the  telescope  is  first 
directed  to  A  and  the  vernier  is 
read;  then  to  B,  C,  D,  E,  etc.,  in 
succession,  the  vernier  being  read 
at  each  pointing.  The  reading 
of  the  vernier  on  A,  subtracted 
from  that  on  B,  will  give  the  angle 
AOB  and  so  on.  It  is  necessary 
in  this  method,  to  read  both  to 
the  right  and  to  the  left,  and  with  the  telescope  both  direct  and 
inverted.  Since  each  angle  is  measured  on  only  one  part  of  the 
limb,  it  is  necessary  after  completing  the  readings  once  around 
and  back,  to  shift  the  vernier  to  another  part  of  the  limb  and 
repeat  the  readings  in  both  directions,  and  with  the  telescope 
direct  and  inverted.  This  is  done  as  many  times  as  there  are  sets 
of  readings.  Each  complete  set  of  readings  to  right  and  left,  with 


206 


PLANE    SURVEYING 


199 


the  telescope  direct  and  inverted,  gives  one  value  for  each  angle. 

The  lengths  of  the  sides  of  the  triangles  should  be  calculated 
with  extreme  accuracy  in  two  ways  if  possible,  and  by  at  least  two 
persons.  Plane  trigonometry  may  be  used  for  even  extensive 
surveys;  for  though  these  sides  are  really  arcs  and  not  straight 
lines  the  error  under  ordinary  circumstances  will  be  inappreciable. 

Radiating  Triangulation.  This  method  as  is  illustrated  in 
Fig.  131  consists  in  choosing  a  conspicuous  point  O,  nearly  in  the 
center  of  the  area  to  be  surveyed.  Other  points  as  A,  B,  C,  D, 
etc.,  are  so  chosen  that  the  signal  at  ()  can  be  seen  from  all  of 
them,  and  that  the  triangles  ABO,  BCO,  etc.,  shall  be  as  nearly 
equilateral  as  possible.  Measure 
one  side,  as  AB  for  example, 
and  at  A  measure  the  angles 
OAB  and  OAG;  at  B  measure 
the  angles  OB  A  and  OBC;  and 
so  on  around  the  polygon.  The 
correctness  of  these  measure- 
ments may  be  tested  by  the  sum 
of  the  angles.  It  will  seldom  be 
the  case,  however,  that  the  sum 
of  the  angles  will  come  out  just 
even,  and  the  angles  must  then 
be  adjusted,  as  will  be  explained 

later.  The  calculations  of  the  lengths  of  the  unknown  sides  are 
readily  made  by  the  usual  trigonometric  methods;  thus  in  the  tri- 
angle AOB,  there  are  given  one  side  and  all  of  the  angles  of  the 
triangle  from  which  to  calculate  AO  and  BO.  Similarly  all  of 
the  triangles  of  the  polygon  may  be  solved,  and  finally  the  length 
of  OA  may  be  measured  and  compared  with  the  calculated  length, 
&p  found  from  the  first  triangle. 

A  farm  or  field  may  be  surveyed  by  the  previously  described 
method,  but  the  following  plan  will  often  be  more  convenient  : 
Choose  a  base  line  as  AB  within  the  field  and  measure  its  length. 
Consider  first  the  triangles  which  have  AB  for  a  base,  and  the 
corners  of  the  field  for  vertices.  In  the  triangle  ACB  for  example 
(see  Fig.  132),  we  measure  the  angles  CAB  and  CBA  and  the 
length  of  the  base  line  AB,  We  can  therefore  calculate  the  length 


207 


200  PLANE    SURVEYING 

of  AC  and  BC.  Next  consider  the  field  as  made  up  of  triangles 
with  a  common  vertex  A.  In  each  of  them,  two  sides  and  the 
included  angle  are  given,  to  find  the  third  side.  If  now  the  point 
B  at  the  other  end  of  the  base  line  be  taken  for  a  common  vertex, 
a  check  w'ill  be  obtained  upon  the  work. 

A  field  or  a  farm  or  any  inaccessible  area  such  as  a  swamp, 
a  lake,  etc.,  may  be  surveyed  without  entering  it.  For  a  farm  or 
any  area  permitting  unobstructed  vision,  it  will  only  be  necessary 

to  choose  a  base  line  AB,  from 
which  all  of  the  corners  of  the 
farm,  or  all  of  the  salient  points 
of  the  area,  can  be  seen.  Take 
their  bearings,  or  the  angles  be- 
tween the  base  line  and  their 
directions.  The  distances  from 
A  and  B  to  each  of  them  can  be 
calculated  as  described,  and  the 
figure  will  then  sho.w  in  what 
manner  the  content  of  the  field 
is  the  difference  between  the  contents  of  the  triangles  having  A 
or  B  for  a  vertex,  which  lie  outside  of  it,  and  those  which  lie 
partly  within  the  field  and  partly  outside  of  it.  Their  contents 
can  be  calculated,  and  their  difference  will  be  the  desired  content. 
See  Fig.  133.  Evidently  the  entire  area  included  between  the  cor- 
ners of  the  field  and  the  base  line  is  the  sum  of  the  triangles  A2B, 
2B3  and  3B4.  Subtracting  from  this  sum  the  areas  of  the  tri- 
angles 2A1,  1AB,  1B6,  56B  and  5B4,  there  will  remain  the 
required  area  of  the  field,  123456. 

In  all  of  the  operations  which  have  been  explained,  the  posi- 
tion of  a  point  has  been  determined  by  taking  the  angles,  or  bear- 
ings, of  two  lines  passing  from  the  two  ends  of  a  base  line  to  the 
unknown  point,  but  the  same  determination  may  be  effected 
inversely  by  taking  from  the  point  the  bearings  by  compass  of  the 
two  ends  of  the  base  line  or  any  two  known  points.  The  unknown 
point  will  then  be  fixed  by  plotting  from  the  two  known  points, 
the  opposite  bearings,  for  it  will  be  at  the  intersection  of  the  lines 
thus  determined. 


208 


PLANE    SURVEYING 


201 


The  determination  of  a  point  by  the  method  founded  on  the 
intersection  of  lines,  has  the  serious  defect  that  the  point  sighted 
to  will  be  very  indefinitely  determined  if  the  lines  which  fix  it 
meet  at  a  very  acute  or  a  very  obtuse  angle,  which  the  relative 
position  of  the  points  observed  from  and  to  often  render  unavoid- 
able. Intersections  at  right 
angles  should  therefore  be 
sought  for,^so  far  as  other  con- 
siderations will  permit. 

Adjusting  the  Triangle. 
All  of  the  angles  of  a  given  tri- 

G  O 

angle  are  measured.     If  but  two 

have    been    measured,   and    the 

third  computed,  the  entire  error 

of  measurement  of  the  two  angles 

will    be   thrown   into  the  third 

angle.     It  will   be  found,  upon 

adding  together  the  measured  angles  of  a  triangle,  that  the  sam  of 

the  three  angles  is  almost  invariably  more  or  less  than  180°.     With 

*->  J 

the  engineer's  transit  the  error  should  be  less  than  one  minute. 
If  there  is  no  reason  to  suppose  that  one  angle  is  measured  more 
carefully  than  another,  this  error  should  be  divided  equally  among 
the  three  angles  of  the  triangle,  and  the  corrected  angles  are  used 
in  computing  the  azimuths  and  lengths  of  the  sides.  This  distri- 
bution of  the  error  is  called  "adjusting"  the  triangle.  With  the 
large  systems  of  extensive  geodetic  surveys  much  more  elaborate 
methods  are  employed,  since  a  large  number  of  triangles  must  be 
adjusted  simultaneously  so  that  they  will  all  be  geometrically  con- 
sistent, not  only  each  by  itself,  but  one  with  another. 


209 


MEDIUM    PRICED     DRAWING    OUTFIT    SHOWING    THE    INSTRUMENTS    NECL&. 
SARY    FOR    PURSUING    A    COURSE    IN    MECHANICAL    DRAWING. 


MECHANICAL  DRAWING 

PART   I 


The  subject  of  mechanical  drawing  is  of  great  interest  and 
importance  to  all  mechanics  and  engineers.  Drawing  is  the 
method  used  to  show  graphically  the  small  details  of  machinery; 
it  is  the  language  by  which  the  designer  speaks  to  the  workman; 
it  is  the  most  graphical  way  to  place  ideas  and  calculations  on 
record.  Working  drawings  take  the  place  of  lengthy  explana- 
tions, either  written  or  verbal.  A  brief  inspection  of  an  accurate, 
well-executed  drawing  gives  a  better  idea  of  a  machine  than  a 
large  amount  of  verbal  description.  The  better  and  more  clearly 
a  drawing  is  made,  the  more  intelligently  the  workman  can  com- 
prehend the  ideas  of  the  designer.  A  thorough  training  in  this 
important-  subject  is  necessary  to  the  success  of  everyone  engaged 
in  mechanical  work.  The  success  of  a  draftsman  depends  to  some 
extent  upon  the  quality  of  his  instruments  and  materials.  Begin- 
ners frequently  purchase  a  cheap  grade  of  instruments.  After 
they  have  become  expert  and  have  learned  to  take  care  of  their 
instruments  they  discard  them  for  those  of  better  construction  and 
finish.  This  plan  has  its  advantages,  but  to  do  the  best  work, 
strong,  well-made  and  finely  finished  instruments  are  necessary. 

INSTRUMENTS  AND  MATERIALS. 

Drawing  Paper.  In  selecting  drawing  paper,  the  first  thing 
to  be  considered  is  the  kind  of  paper  most  suitable  for  the  pro- 
posed work.  For  shop  drawings,  a  manilla  paper  is  frequently 
used,  on  account  of  its  toughness  and  strength,  because  the  draw- 
ing is  likely  to  be  subjected  to  considerable  hard  usage.  If  a 
finished  drawing  is  to  be  made,  the  best  white  drawing  paper 
should  be  obtained,  so  that  the  drawing  will  not  fade  or  become 
discolored  with  age.  A  good  drawing  paper  should  be  strong, 
have  uniform  thickness  and  surface,  should  stretch  evenly,  and 
should  neither  repel  nor  absorb  liquids.  It  should  also  allow  con- 
siderable erasing  without  spoiling  the  surface,  and  it  should  lie 
smooth  when  stretched  or  when  ink  or  colors  are  used.  It  is,  of 

Copyright,  1'JOU,  by  American  School  of  Correspondence. 


211 


MECHANICAL  DRAWING 


course,  impossible  to  find  all  of  these  qualities  in  any  one  paper, 
as  for  instance  great  strength  cannot  be  combined  with  fine 
surface. 

In  selecting  a  drawing  paper  the  kind  should  be  chosen 
which  combines  the  greatest  number  of  these  qualities  for  the 
given  work.  Of  the  better  class  Whatman's  are  considered  by 
far  the  best.  This  paper  is  made  in  three  grades;  the  hoi 
pressed  has  a  smooth  surface  and  is  especially  adapted  for  pencil 
and  very  fine  line  drawing,  the  cold  pressed  is  rougher  than 
the  hot  pressed,  has  a  finely  grained  surface  and  is  more  suit- 
able for  water  color  drawing;  the  rough  is  used  for  tinting.  The 
cold  pressed  does  not  take  ink  as  well  as  the  hot  pressed,  but 
erasures  do  not  show  as  much  on  it,  and  it  is  better  for  general 
work.  There  is  but  little  difference  in  the  two  sides  of  Whatman's 
paper,  and  either  can  be  used.  This  paper  comes  in  sheets  of 
standard  sizes  as  follows: — 

Cap,  13X17  inches.      Elephant,  23  X  28  inches 

Demy,  15  X  20 

Medium,  17  X  22 

Royal,  19  X  24 

Super-Royal,  19  X  27 

Imperial,  22  X  30 


lleph 
'ohm 


Columbia,  23  X  34 

Atlas,  26  X  34 

Double  Elephant,  27  X  40 

Antiquarian,  31  X  53 

Emperor,  48  X  68 


The  usual  method  of  fastening  paper  to  a  drawing  board  is  by 
means  of  thumb  tacks  or  small  one-ounce  copper  or  iron  tacks. 
In  fastening  the  paper  by  this  method  first  fasten  the  upper  left 
hand  corner  and  then  the  lower  right  pulling  the  paper  taut.  The 
other  two  corners  are  then  fastened,  and  sufficient  number  of  tacks 
are  placed  along  the  edges  to  make  the  paper  lie  smoothly.  For 
very  fine  work  the  paper  is  usually  stretched  and  glued  to  the 
board.  To  do  this  the  edges  of  the  paper  are  first  turned  up  all 
the  way  round,  the  margin  being  at  least  one  inch.  The  whole 
surface  of  the  paper  included  between  these  turned  up  edges  is 
then  moistened  by  means  of  a  sponge  or  soft  cloth  and  paste  or 
glue  is  spread  on  the  turned  up  edges.  After  removing  all  the 
surplus  water  on  the  paper,  the  edges  are  pressed  down  on  the 
board,  commencing  at  one  corner.  During  this  process  of  laying 
down  the  edges,  the  paper  should  be  stretched  slightly  by  pulling 
the  edges  towards  the  edges  of  the  drawing  board.  The  drawing 
board  is  then  placed  horizontally  and  left  to  dry.  After  the  paper 
has  become  dry  it  will  be  found  to  be  as  smooth  and  tight  as  a 


212 


MECHANICAL  DRAWING 


drum  head.     If,  in  stretching,  the  paper  is  stretched  too  much  it 
is  likely  to  split  in  drying.     A  sliglit  stretch  is  sufficient. 

Drawing  Board.  The  size  of  the  drawing  board  depends 
upon  the  size  of  paper.  Many  draftsmen,  however,  have  several 
boards  of  various  sizes,  as  they  are  very  convenient.  The  draw- 
ing board  is  usually  made  of  soft  pine,  which  should  be  well  sea- 
soned and  straight  grained.  The  grain  should  run  lengthwise  of 
the  board,  and  at  the  two  ends  there  should  be  pieces  about  If  or 
2  inches  wide  fastened  to  the  board  by  nails  or  screws.  These 
end  pieces  should  be  perfectly  straight  for  accuracy  in  using  the 
T-square.  Frequently  the  end  pieces  are  fastened  by  a  glued 


DRAWING  BOARD 

matched  joint,  nails  and  screws  being  also  used.  Two  cleats  on 
the  bottom  extending  the  whole  width  of  the  board,  will  reduce 
the  tendency  to  warp,  and  make  the  board  easier  to  move  as  they 
raise  it  from  the  table. 

Thumb  Tacks.  Thumb  tacks  are  used  for  fastening  the 
paper  to  the  drawing  board.  They  are  usually  made  of  steel 
either  pressed  into  shape,  as  in  the  cheaper  grades,  or  made  with  a 
head  of  German  silver  with  the  point  screwed  and  riveted  to  it. 
They  are  made  in  various  sizes  and  are  very  convenient  as  they 
can  be  easily  removed  from  the  board.  For  most  work  however, 


213 


MECHANICAL  DRAWING 


draftsmen  use  small  one-ounce  copper  or  iron  tacks,  as  they  can  be 
forced  flush  with  the  drawing  paper,  thus  offering  no  obstruction 
to  the  T-square.    They  also  possess  the  advantage  of  cheapness. 
Pencils.     In   pencilling  a   drawing   the   lines   should   be   very 
fine  and  light.     To  obtain  these  light  lines  a  hard  lead  pencil  must 
be  used.     Lead   pencils  are  graded   according  to   their  hardness, 
and  are  numbered  by  using  the  letter  H.     In  general  a  lead  pencil 
of  5H  (or  HHHHH)  or  6H  should  be  used.    A  softer  pencil,  4H, 
is  better   for    making  letters,     figures  and 
points.      A   hard    lead    pencil   should    be 
sharpened  as  shown  in  Fig.  1.     The  -wood 
is    cut  away  so    that  about  1  or  ^  inch 
of    lead  projects.     The  lead  can  then  be 
sharpened  to  a  chisel  edge   by  rubbing  it 
against  a  bit  of   sand  paper  or   a  fine  file. 
It  should  be  ground   to  a  chisel   edge  and 
the  corners  slightly  rounded.       In  making 
the   straight    lines   the  chisel   edge   should 
be  used  by  placing  it  against  the  T-square 
or  triangle,  and  because  of  the  chisel  edge 

the  lead  will  remain  sharp  much  longer  than  if  sharpened  to  a  point. 
This  chisel  edge  enables  the  draftsman  to  draw  a  fine  line  exactly 
through  a  given  point.  If  the  drawing  is  not  to  be  inked,  but  is 
made  for  tracing  or  for  rough  usage  in  the  shop,  a  softer  pencil, 
3H  or  4H,  may  be  used,  as  the  lines  will  then  be  somewhat  thicker 
and  heavier.  The  lead  for  compasses  may  also  be  sharpened  to  a 
point  although  some  draftsmen  prefer  to  use  a  chisel  edge  in  the 
compasses  as  well  as  for  the  pencil. 

In  using  a  very  hard  lead  pencil,  the  chisel  edge  will  make  a 
deep  depression  in  the  paper  if  much  pressure  is  put  on  the  pencil. 
As  this  depression  cannot  be  erased  it  is  much  better  to  press 
lightly  on  the  pencil. 

Erasers.  In  making  drawings,  but  little  erasing  should  be 
necessary.  However,  in  case  this  is  necessary,  a  soft  rubber 
should  be  used.  In  erasing  a  line  or  letter,  great  care  must  be 
exercised  or  the  surrounding  work  will  also  become  erased.  To 
prevent  this,  some  draftsmen  cut  a  slit  about  3  inches  long  and 
4  to  £  inch  wide  in  a  card  as  shown  in  Fig.  2.  The  card  is  then 


214 


MECHANICAL  DRAWING 


placed  over  the  work  and  the  line  erased  without  erasing  the  rest 
of  the  drawing.  An  erasing  shield  of  a  form  similar  to  that  shown 
in  Fig.  3  is  very  convenient,  especially  in  erasing  letters.  It  is 
made  of  thin  sheet  metal  and  is  clean  and  durable. 

For  cleaning  drawings,  a  sponge  rubber  may  be  used.     Bread 
crumbs  are  also  used   for  this  purpose.      To  clean   the  drawing 


o 

0 


o 

o 


big.  2. 


Fig. 


scatter  dry  bread  crumbs  over  it  and  rub  them  on  the  surface 
with  the  hand. 

T-Square.    The    T-square    consists    of    a    thin    straight    edge 
called  the  blade,  fastened  to  a  head  at  right  angles  to  it.     It  gets 


Fig.  4. 

its  name  from  the  general  shape.  T-squares  are  made  of  various 
materials,  wood  being  the  most  commonly  used.  Fig.  4  shows  an 
ordinary  form  of  ri '-square  which  is  adapted  to  most  work.  In 
Fig.  5  is  shown  a  T-square  with  edges  made  of  ebony  or  mahogany, 
as  these  woods  are  much  harder  than  pear  wood  or  maple,  which 
is  generally  used.  The  head  is  formed  so  as  to  fit  against  the  left- 
hand  edge  of  the  drawing  board,  while  the  blade  extends  over  the 
surface.  It  is  desirable  to  have  the  blade  of  the  T-square  form  a 
right  angle  with  the  head,  so  that  the  lines  drawn  with  the  T-- 
square will  be  at  right  angles  to  the  left-hand  edge  of  the  board. 
This,  however,  is  not  absolutely  necessary,  because  the  lines  drawn 
with  the  T-square  are  always  with  reference  to  one  edge  of  the 


215 


MECHANICAL    DRAWING. 


board  only,  and  if  this  edge  of  the  board  is  straight,  the  lines 
drawn  with  the  T-square  will  be  parallel  to  each  other.  The  T- 
square  should  never  be  used  except  with  the  left-hand  edge  of  the 
board,  as  it  is  almost  impossible  to  find  a  drawing  broad  with  the 
edges  parallel  or  at  right  angles  to  each  other. 

The  T-square  with  an  adjustable  head  is  frequently  veiy  con- 
venient, as  it  is  sometimes  necessary  to  draw  lines  parallel  to  each 


Fig.  5. 

other  which  are  not  at  right  angles  to  the  left-hand  edge  of  the 
board.  This  form  of  T-square  is  similar  to  the  ordinary  T-square 
already  described,  but  the  head  is  swiveled  so  that  it  may  be 
clamped  at  any  desired  angle.  The  ordinary  T-square  as  shown 

in  Figs.  4  and  5  is,  how 
ever,  adapted  to  almost 
any  class  of  drawing. 

*  Fig.  6  shows  the 
method  of  drawing  parallel 
horizontal  lines  witli  the 
T-square.  With  the  head 
of  the  T-square  in  contact 
with  the  left-hand  edge  of 
the  board,  the  lines  may  be 

drawn  by  moving  the  T-square  to  the  desired  position.  In  using  the 
T-square  the  upper  edge  should  always  be  used  for  drawing  as  the 
two  edges  may  not  be  exactly  parallel  and  straight,  and  also  it  is 
more  convenient  to  use  this  edge  with  the  triangles.  If  it  is  neces- 
sary to  use  a  straight  edge  for  trimming  drawings  or  cutting  the 
paper  from  the  board,  the  lower  edge  of  the  T-square  should  be 
used  so  that  the  upper  edge  may  not  be  marred. 

For  accurate  work  it  is  absolutely  necessary  that  the  working 
edge  of  the   T-square   should  be  exactly   straight.     To  test  the 


216 


MECHANICAL    DRAWING. 


straightness  of  the  edge  of  the  T-square,  two  T-squares  may  be 
placed  together  as  shown  in  Fig.  7.  This  figure  shows  plainly 
that  the  edge  of  one  of  the  T-squares  is  crooked.  This  fact,  how- 
ever, does  not  prove  that  either  one  is  straight,  and  for  this  deter- 
mination a  third  blade  must  be 
used  and  tried  with  the  two 
given  T-squares  successively. 

Triangles.  Triangles  are 
made  of  various  substances  such 
as  wood,  rubber,  celluloid  and 
steel.  Wooden  triangles  are 
cheap  but  are  likely  to  warp  and  get  out  of  shape.  The  rubber  tri- 
angles are  frequently  used,  and  are  in  general  satisfactory.  The 
transparent  celluloid  triangle  is,  however,  extensively  used  on  ac- 
count' of  its  transparency,  which  enables  the  draftsmen  to  see  the 
work  already  done  even  when  covered  with  the  triangle.  In  using 
a  rubber  or  celluloid  triangle  take  care  that  it  lies  perfectly  flat  or 


Fig.  7. 


TRIANGLES. 


is  hung  up  when  not  in  use ;  when  allowed  to  lie  on  the  drawing 
board  with  a  pencil  or  an  eraser  under  one  corner  it  will  become 
warped  in  a  short  time,  especially  if  the  room  is  hot  or  the  sun 
happens  to  strike  the  triangle. 

Triangles  are  made  in  various  sizes,  and  many  draftsmen 
have  several  constantly  on  hand.  A  triangle  from  6  to  8  inches 
on  a  side  will  be  found  convenient  for  most  work,  although  there 
are  many  cases  where  a  small  triangle  measuring  about  4  inches 


217 


10 


MECHANICAL    DRAWIKO. 


on  a  side  will  be  found  useful.  Two  triangles  are  necessary  for 
every  draftsman,  one  having  two  angles  of  45  degrees  each  and 
one  a  right  angle  ;  and  the  other  having  one  angle  of  60  degrees, 
one  of  30  degrees  and  one  of  90  degrees. 

The  value  of  the  triangle  depends  upon  the  accuracy  of  the 
angles  and  the  straightness  of  the  edges.     To  test  the  accuracy  of 

the  right  angle  of  a  tri- 
angle, place  the  triangle 
with  the  lower  edge  rest- 
ing on  the  edge  of  the 
T-square,  as  shown  in 
Fig.  8.  Now  draw  the 
line  C  D,  which  should  be 
perpendicular  to  the  edge 
of  the  T-square.  The 
same  triangle  should -then 
be  placed  in  the  position  shown  at  B.  If  the  right  angle  of  the 
triangle  is  exactly  90  degrees  the  left-hand  edge  of  the  triangle 
should  exactly  coincide  with  the  line  C  D. 

To  test  the  accuracy  of  the  45-degree  triangles,  first  test  the 
right  angle  then  place  the 
triangle  with  the  lower 
edge  resting  on  the  work- 
ing edge  of  the  T-square, 
and  draw  the  line  E  F  as 
shown  in  Fig.  9.  Now 
without  moving  the  T-- 
square place  the  triangle 


Fig.  8. 


Fig.  9. 


so  that  the  other  45-degree 

angle    is    in    the  position 

occupied  by  the  first.     If  the  two  45-degree  angles  coincide  they 

are  accurate.  • 

Triangles  are  very  convenient  in  drawing  lines  at  right 
angles  to  the  T-square.  The  method  of  doing  this  is  shown  in 
Fig.  10.  Triangles  are  also  used  in  drawing  lines  at  an  angle 
with  the  horizontal,  by  placing  them  on  the  board  as  shown  in 
Fig.  11.  Suppose  the  line  E  F  (Fig.  12)  is  drawn  at  any  anjle, 
and  we  wish  to  draw  a  line  through  the  point  P  parallel  to  \i 


218 


MECHANICAL     DRAWING. 


li 


First  place  one  of  the  triangles  as  shown  at  A,  having  one  edge 
coincidkg  with  the  given  line.  Now  take  the  other  triangle  and 
place  one  of  its  edges  in  contact  with  the  bottom  edge  of  triangle 
A.  Holding  the  triangle  B  firmly  with  the  left  hand  the  triangle 
A  may  be  slipped  along  to  the  right  or  to  the  left  until  the  edge 
of  the  triangle  reaches  the 
point  P.  The  line  M  N 
may  then  be  drawn  along 
the  edge  of  the  triangle 
passing  through  the  point 
P.  In  place  of  the  tri- 
angle B  any  straight  edge 
such  as  a  T-square  may  be 
used. 

A  line  can  be  drawn 
perpendicular  to  another  by  means  of  the  triangles  as  follows. 
Let  E  F  (Fig.  13)  be  the  given  line,  and  suppose  we  wish  tc 
draw  a  line  perpendicular  to  E  F  through  the  point  D.  Place 
the  longest  side  of  one  of  the  triangles  so  that  it  coincides 

with  the  lina  E  F,  as  the 
triangle  is  snown  in.  posi- 
tion at  A.  Place  the  other 
triangle  (or  any  straight 
edge)  in  the  position  ot 
the  triangle  as  shown  at 
B,  one  edge  resting  against 
the  edge  of  the  triangle  A. 


Fig,  10. 


Fig.  11. 


Then  holding  B  with  the 
left   hand,    place    the    tn 

angle  A  in  the  position  shown  at  C,  so  that   the   longest   side 

passes  through  the  point  D.     A  line  can  then  be  drawn  through 

the  point  D  perpendicular  to  E  F. 

In  previous  figures  we  have  seen  how  lines  may  be  drawn 

making  angles  of  30,  45,  60  and  £0  degrees  with  the  horizontal. 

If  it  is  desired  to  draw  lines  forming  angles  of  15  and  75  degrees 

the  triangles  may  be  placed  as  shown  in  Fig.  14. 

In  using  the  triangles  and  T-square  almost  any  line  may  b« 

drawn.     Suppose  we  wish  to  draw  a  rectangle  having  one  side 


219 


i'j 


MECHANICAL    DRAWING. 


horizontal.  First  place  the  T-square  as  shown  in  Fig.  15.  By 
moving  the  T-square  up  or  down,  the  sides  A  B  and  D  C  may  be 
drawn,  because  they  are  horizontal  and  parallel.  Now  place  one 
of  the  triangles  resting  on  the  T-square  as  shown  at  E,  and  hav- 
ing the  left-hand  edge  passing  through  the  poirt  D.  The  vertical 


Fig.  12. 


Fig.  13. 


line  D  A  may  be  drawn,  and  by  sliding  the  triangle  along  the  edge 
of  tlie  T-square  to  the  position  F  the  line  B  C  may  be  drawn  by 
using  the  same  edge.  These  positions  are  shown  dotted  in  Fig.  15. 
If  the  rectangle  is  to  be  placed  in  some  other  position  on  the 
drawing  board,  as  shown  in  Fig.  16,  place  the  45-degree  triangle 

F  so  that  one  edge  is 
parallel  to  or  coincides 
with  the  side  D  C.  Now 
holding  the  triangle  F  in 
position  place  the  triangle 
H  so  that  its  upper  edge 
coincides  with  the  lower 
edge  of  the  triangle  F. 
By  holding  H  in  position 
and  sliding  the  triangle  F 
along  its  upper  edge,  the  sides  A  B  and  D  C  may  be  drawn. 
To  draw  the  sides  A  D  and  B  C  the  triangle  should  be  used  as 
shown  at  E. 

Compasses.  Compasses  are  used  for  drawing  circles  and 
arcs  of  circles.  They  are  made  of  various  materials  and  in  various 
sizes.  The  cheaper  class  of  instruments  are  made  of  brass,  but 
they  are  unsatisfactory  on  account  of  the  odor  and  the  tendency 
to  tarnish  The  best  material  is  German  silver.  It  does  not  soil 


L 


Fig.  14. 


220 


MECHANICAL    DRAWING. 


readily,  it  has  no  odor,  and  is  easy  to  keep  clean.  Aluminum  in- 
struments possess  the  advantage  of  lightness,  but  on  account  of 
the  soft  metal  they  do  not  wear  well. 

The  compasses  are  made  in  the  form  shown  in  Figs.  17  and 
18.  Pencil  and  pen  points  are  provided,  as  shown  in  Fig.  17. 
Either  pen  or  pencil  may  be  inserted  in  one  leg  by  menns  of  a 
shank  and  socket.  The 
other  leg  is  fitted  with  a 
needle  point  which  is 
placed  at  the  center  of  the 
circle.  In  most  instru- 
ments the  needle  point  is 
separate,  and  is  made  of  a 
piece  of  round  steel  wire 
having  a  square  shoulder 
at  one  or  both  ends.  Be- 


r— 

0 
A* 

o 

K 
hK 

I 

."»                                                                  ° 

o 

o 

Fig.  15. 


low  this  shoulder  the  needle  point  projects.  The  needle  is 
made  in  this  form  so  that  the  hole  in  the  paper  may  be  very 
minute. 

In  some  instruments  lock  nuts  are  used  to    hold   the  joint 
firmly  in  position.     These  lock  nuts  are  thin  discs  of  steel,  with 

notches  for  using  a  wrench  or 
forked  key.  Fig.  19  shows  the 
detail  of  the  joint  of  high  grade 
instruments.  Both  legs  are  alike 
at  the  joint,  and  two  pivoted 
screws  are  inserted  in  the  yoke. 
This  permits  ample  movement 
of  the  legs,  and  at  the  same 
time  gives  the  proper  stiff- 
ness. The  flat  surface  of  one  of 
the  legs  is  faced  with  steel,  the  other  being  of  German  silver, 
in  order  that  the  rubbing  parts  may  be  of  different  metals.  Small 
set  screws  are  used  to  prevent  the  pivoted  screws  from  turning 
in  the  yoke.  The  contact  surfaces  of  this  joint  are  made  cir- 
cular to  exclude  dust  and  dirt  and  to  prevent  rusting  of  the 
steel  face. 

Figs.  20,  21  and   22  show  the  detail  of  the  socket;   in   some 


Fig.  16. 


MECHANICAL    DRAWING. 


instruments  the  shank  and  socket  are  pentagonal,  as  shown  in 
Fig.  20.  The  shank  enters  the  socket  loosely,  and  is  held  in  place 
by  means  of  the  screw.  Unless  used  very  carefully  this  arrange- 
ment is  not  durable  because  the  sharp  corners  soon  wear,  and  the 
pressure  on  the  set  screw  is  not  sufficient  to  hold  the  shank  firmly 
in  place. 

In  Fig.  21  is  shown  another  form  of  shank.  This  is  round, 
having  a  flat  top.  A  set  screw  is  also  used  to  hold  this  in  posi- 
tion. A  still  better  form  of  socket  is  shown  in  Fig.  22 ;  the  hole 


Fig.  17. 


Fig.  18. 


is  made  tapered  and  is  circular.  The  shank  fits  accurately,  and 
is  held  in  perfect  alignment  by  a  small  steel  key.  The  clamping 
screw  is  placed  upon  the  side,  and  keeps  the  two  portions  of  the 
split  socket  together. 

Figs.  17  and  18  show  that  both  legs  of  the  compasses  are 
jointed  in  order  that  the  lower  part  of  the  legs  may  be  perpen- 
dicular to  the  paper  while  drawing  circles.  In  this  way  the 
ueedle  point  makes  but  a  small  hole  in  the  paper,  and  both  nibs  of 


MECHANICAL    DRAWING. 


the  pen  will  press  equally  on  the  paper.  In  pencilling  circles  it 
is  not  as  necessary  that  the  pencil  should  be  kept  vertical ;  it  is  a 
good  plan,  however,  to  learn  to  use  them  in  this  way  both  in  pen- 
cilling and  inking.  The  com- 
passes should  be  held  loosely  be- 
tween the  thumb  and  forefinger. 
If  the  needle  point  is  sharp,  as 
it  should  be,  only  a  slight  pres- 
sure will  be  required  to  keep  it 
in  place.  While  drawing  the 
circle,  incline  the  compasses 
slightly  in  the  direction  of 
revolution  and  press  lightly  on 
the  pencil  or  pen. 

In  removing  the  pencil  or 
pen,    it    should  be   pulled    out  Fi£-  19- 

straight.  If  bent  from  side  to  side  the  socket  will  become  en- 
larged and  the  shank  worn;  this  will  render  the  instrument  inac- 
curate. For  drawing  large  circles  the  lengthening  bar  shown  in 
Fig.  17  should  be  used.  When  using  the  lengthening  bar  the 


Fig.  20. 


Q 

Fig.  21. 
one  hand  and  the  circle 


needle   point  should  be    steadied  with 
described  with  the  other. 

Dividers.     Dividers,  shown  in  Fig.  23,  are  made  similar  to  the 
compasses.     They  are  used  for  laying  off  distances  on  the  draw- 
ing, either  from  scales  or  from  other  parts  of  the  drawing.     They 
_  may  also  be  used  for  dividing  a  line 

( 1 1  .       V-^_ i  into   equal  parts.      When  dividing  a 

line  into  equal  parts  the  dividers 
should  be  turned  in  the  opposite  direc- 
tion each  time,  so  that  the  moving  point  passes  alternately  to 
the  right  and  to  the  left.  The  instrument  can  then  be  operated 
readily  with  one  hand.  The  points  of  the  dividers  should  be 
very  sharp  so  that  the  holes  made  in  the  paper  will  be  small 
If  large  holes  are  made  in  the  paper,  and  the  distances  betweer 


Fig.  22. 


223 


16  MECHANICAL    DRAWING. 

the  points  are  not  exact,  accurate  spacing  cannot  be  done 
Sometimes  the  compasses  are  furnished  with  steel  divider  points 
in  addition  to  the  pen  and  pencil  points.  The  compasses  may 
then  be  used  either  as  dividers  or  as  compasses.  Many  drafts- 
men use  a  needle  point  in  place  of  dividers  for  making  measure- 
ments from  a  scale.  The  eye  end  of  a  needle  is  first  broken  off 
and  the  needle  then  forced  into  a  small  handle  made  of  a  round 
piece' of  soft  pine.  This  instrument  is  very  convenient 
for  indicating  the  intersection  of  lines  and  marking  off 
distances. 

Bow  Pen  and  Bow  Pencil.  Ordinary  large  compasses 
are  too  heavy  to  use  in  making  small  circles,  fillets,  eta 
The  leverage  of  the  long  leg  is  so  great  that  it  is  very 
difficult  to  draw  small  circles  accurately.  For  this  reason 
the  bow  compasses  shown  in  Figs.  24  and  25  should  be 
used  on  all  arcs  and  circles  having  a  radius  of  less  than 
three-quarters  inch.  The  bow  compasses  are  also  con- 
venient for  duplicating  small  circles  such  as  those  which 
represent  boiler  tubes,  bolt  holes,  etc.,  «ince  there  is  no 
tendency  to  slip. 

The  needle  point  must  be  adjusted  to  the  same 
length  as  the  pen  or  pencil  point  if  very  small  circles  are 
to  be  drawn.  The  adjustment  for  altering  the  radius  of 
the  circle  can  be  made  by  turning  the  nut.  If  the  change 
in  radius  is  considerable  the  points  should  be  pressed  to- 
gether to  remove  the  pressure  from  the  nut  which  can 
Pig.  23  tnen  ke  turned  in  either  direction  with  but  little  wear  on 

the  threads. 

Fig.  26  shows  another  bow  instrument  which  is  frequently 
used  in  small  work  in  place  of  the  dividers.  It  has  the  advantage 
of  retaining  the  adjustment. 

Drawing  Pen.  For  drawing  straight  lines  and  curves  that 
are  not  arcs  of  circles,  the  line  pen  (sometimes  called  the  ruling 
pen)  is  used.  It  consists  of  two  blades  of  steel  fastened  to  a 
handle  as  shown  in  Fig.  27.  The  distance  between  the  pen  points 
can  be  adjusted  by  the  thumb  screw,  thus  regulating  the  width  of 
line  to  be  drawn.  The  blades  are  given  a  slight  curvature  so  that 
there  will  be  a  cavity  for  ink  when  the  points  are  close  together. 


MECHANICAL    DRAWING. 


17 


The  pen  may  be  filled  by  means  of  a  common  steel  pen  or 
with  the  quill  which  is  provided  with  some  liquid  inks.  The  pen 
should  not  be  dipped  in  the  ink  because  it  will  then  be  necessary 
to  wipe  the  outside  of  the  blades  before  use.  The  ink  should 
fill  the  pen  to  a  height  of  about  £  or  |  inch ;  if  too  much  ink  is 
placed  in  the  pen  it  is  likely  to  drop  out  and  spoil  the  drawing. 
Upon  finishing  the  work  the  pen  should  be  carefully  wiped  with 


Fig.  24. 


Fig.  25. 


Fig.  26. 


jhamois  or  a  soft  cloth,  because  most  liquid  inl.s  corrode  the  steel. 
In  using  the  pen,  care  should  be  taken  that  both  blades  bear 
equally  on  the  paper.  If  the  points  do  not  bear  equally  the  line 
will  be  ragged.  If  both  points  touch,  and  the  pen  is  in  good 
condition  the  line  will  be  smooth.  The  pen  is  usually  inclined 
slightly  in  the  direction  in  which  the  line  is  drawn.  The  pen 


Fig.  27. 

should  tounh  the  triangle  or  T-square  which  serve  as  guides,  but 
it  should  not  be  pressed  against  them  because  the  lines  will  then 
be  uneven.  The  points  of  the  pen  should  be  close  to  the  edge  of 
the  triangle  or  T-square,  but  should  not  touch  it. 

To    Sharpen  the  Drawing  Pen.       After  the  pen  has  been 
used  for  some  time  the  points  become  worn,  and  it  is  impossible 


225 


18  MECHANICAL    DRAWING. 

to  make  smooth  lines.  This  is  especially  true  if  rough  paper  is 
used.  The  pen  can  be  put  in  proper  condition  by  sharpening  it. 
To  do  this  take  a  small,  flat,  close-grained  oil-stone.  The  blades 
should  first  be  screwed  together,  and  the  points  of  the  pen  can  be 
given  the  proper  shape  by  drawing  the  pen  back  and  forth  over 
the  stone  changing  the  inclination  so  that  the  shape  of  the  ends 
will  be  parabolic.  This  process  dulls  the  points  but  gives  them 
the  proper  shape,  and  makes  them  of  the  same  length. 

To  sharpen  the  pen,  separate  the  points  slightly  and  rub  one 
of  them  on  the  oil-stone.  While  doing  this  keep  the  pen  at  an 
angle  of  from  10  to  15  degrees  with  the  face  of  the  stone,  and 
give  it  a  slight  twisting  movement.  This  part  of  the  operation 
requires  great  care  as  the  shape  of  the  ends  must  not  be  altered. 
After  the  pen  point  has  become  fairly  sharp  the  other  point 
should  be  ground  in  the  same  manner.  All  the  grinding  should 
be  done  on  the  outside  of  the  blades.  The  burr  should  be 
removed  from  the  inside  of  the  blades  by  using  a  piece  of  leather 
or  a  piece  of  pine  wood. 

Ink  should  now  be  placed  between  the  blades  and  the  pen 
tried.  The  pen  should  make  a  smooth  line  whether  fine  or 
heavy,  but  if  it  does  not  the  grinding  must  be  continued  and  the 
pen  tried  frequently. 

Ink.  India  ink  is  always  used  for  drawing  as  it  makes  a 
permanent  black  line.  It  may  be  purchased  in  solid  stick  form 
or  as  a  liquid.  The  liquid  form  is  very  convenient  as  much  time 
is  saved,  and  all  the  lines  will  be  of  the  same  color;  the  acid  in 
the  ink,  however,  corrodes  steel  and  makes  it  necessary  to  keep 
the  pen  perfectly  clean. 

Some  draftsmen  prefer  to  use  the  India  ink  which  comes  in 
stick  form.  To  prepare  it  for  use,  a  little  water  should  be  placed 
in  a  saucer  and  one  end  of  the  stick  placed  in  it.  The  ink  is 
ground  by  giving  it  a  twisting  movement.  When  the  water  has 
become  black  and  slightly  thickened,  it  should  be  tried.  A 
heavy  line  should  be  made  on  a  sheet  of  paper  and  allowed  to 
dry.  If  the  line  has  a  grayish  appearance,  more  grinding  is 
necessary.  After  the  ink  is  thick  enough  to  make  a  good  black 
line,  the  grinding  should  cease,  because  very  thick  ink  will  not 
flow  freely  from  the  pen.  If,  however,  the  ink  has  become  too 


226 


MECHANICAL    DRAWING.  16 

thick,  it  may  be  diluted  with  water.  After  using,  the  stick 
should  be  wiped  dry  to  prevent  crumbling.  It  is  well  to  grind 
the  ink  in  small  quantities  as  it  does  not  dissolve  readily  if  it  has 
once  become  dry.  If  the  ink  is  kept  covered  it  will  keep  for  two 
or  three  days. 

Scales.  Scales  are  used  for  obtaining  the  various  measure- 
ments on  drawings.  They  are  made  in  several  forms,  the  most 
convenient  being  the  flat  with  beveled  edges  and  the  triangular. 
The  scale  is  usually  a  little  over  12  inches  long  and  is  graduated 
for  a  distance  of  12  inches.  The  triangular  scale  shown  in  Fig. 
28  has  six  surfaces  for  graduations,  thus  allowing  many  gradua- 
tions on  the  same  scale. 

The  graduations  on  the  scales  are  arranged  so  that  the 
drawings  may  be  made  in  any  proportion  to  the  actual  size.  For 
mechanical  work,  the  common  divisions  are  multiples  of  two. 


Thus  we  make  drawings  full  size,  half  size,  J,  £,  Jg,  gL,  g^,  etc. 
If  a  drawing  is  ^  size,  3  inches  equals  1  foot,  hence  3  inches  is 
divided  into  12  equal  parts  and  each  division  represents  one  inch. 
If  the  smallest  division  on  a  scale  represents  Jg  inch,  the  scale  is 
said  to  read  to  Jg  inch. 

Scales  are  often  divided  into  y1^,  £•$,  ^,  3^,  etc.,  for  archi- 
tects, civil  engineers,  and  for  measuring  on  indicator  cards. 

The  scale  should  never  be  used  for  drawing  lines  in  place  of 
triangles  or  T-square. 

Protractor.  The  protractor  is  an  instrument  used  for  laying 
off  and  measuring  angles.  It  is  made  of  steel,  brass,  horn  and 
paper.  If  made  of  metal  the  central  portion  is  cut  out  as  shown 
in  Fig.  29,  so  that  the  draftsman  can  see  the  drawing.  The 
outer  edge  is  divided  into  degrees  and  tenths  of  degrees.  Some- 
times the  graduations  are  very  fine.  In  using  a  protractor  a  very 
sharp  hard  pencil  should  be  used  so  that  the  lines  will  be  fine 
and  accurate. 

The  protractor  should  be  placed  so  that  the  given  line  (  pro 


227 


MECHANICAL    DRAWING. 


duced  if  necessary )  coincides  with  the  two  O  marks.  The 
center  of  the  circle  being  placed  at  the  point  through  which  the 
desired  line  is  to  be  drawn.  The  division  can  then  be  marked 
with  the  pencil  point  or  needle  point. 

Irregular  Curve,     One  of  the  conveniences  of  a  draftsman's 


Fig.  29. 

outfit  is  the  French  or  irregular  curve.  It  is  made  of  wood, 
hard  rubber  or  celluloid,  the  last  named  material  being  the  best. 
It  is  made  in  various  shapes,  two  of  the  most  common  being 


Fig.  30. 

shown  in  Fig.  30.  This  instrument  is  used  for  drawing  curves 
other  than  arcs  of  circles,  and  both  pencil  and  line  pen  can  be 
used. 

To  draw  the  curve,  a  series  of  points  is  first  located  and 
then  the  curve  drawn  passing  through  them  by  using  the  part  of 
the  irregular  curve  that  passes  through  several  of  them  The 


MECHANICAL    DRAWING. 


2] 


curve  is  shifted  for  this  work  from  one  position  to  another.  It 
frequently  facilitates  the  work  and  improves  its  appearance  to 
draw  a  free  hand  pencil  curve  through  the,  points  and  then  use  the 
irregular  curve,  talcing  care  that  it  always  fits  at  least  three  points. 
In  inking  the  curve,  the  blades  of  the  pen  must  be  kept 


Fig.  31. 

tangent    to  the  curve,  thus  necessitating  a  continual    change  of 
direction. 

Beam  Compasses.  The  ordinary  compasses  are  not  large 
enough  to  draw  circles  having  a  diameter  greater  than  about  8  or 
10  inches.  A  convenient  instrument  for  larger  circles  is  found 
in  the  beam  compasses  shown  in  Fig.  31.  The  two  parts  called 
channels  carrying  the  pen  or  pencil  and  the  needle  point  are 
clamped  to  a  wooden  beam  ;  the  distance  between  them  being 
equal  to  the  radius  of  the  circle.  Accurate  adjustment  is  obtained 
by  means  of  a  thumb  nut  underneath  one  of  the  channel  pieces. 

LETTERING. 

No  mechanical  drawing  is  finished  unless  all  headings,  titles 
and  dimensions  are  lettered  in  plain,  neat  type.  Many  drawings 
are  accurate,  well-planned  and  finely  executed  but  do  not  present 
a  good  appearance  because  the  draftsman  did  not  think  it  worth 
while  to  letter  well.  Lettering  requires  time  and  patience; 
and  if  one  wishes  to  letter  rapidly  and  well  he  must  practice. 

Usually  a  beginner  cannot  letter  well,  and  in  order  to  pro 
duce  a  satisfactory  result,  considerable  practice  is  necessary.  Many 


MECHANICAL    DRAWING. 


think  it  a  good  plan  to  practice  lettering  before  commencing  a 
drawing.  A  good  writer  does  not  always  letter  well  ;  a  poor 
writer  need  not  be  discouraged  and  think  he  can  never  learn  to 
make  a  neatly  lettered  drawing. 

In  making  large  letters  for  titles  and  headings  it  is  often 
necessary  to  use  drawing  instruments  and  mechanical  aids.  The 
small  letters,  such  as  those  used  for  dimensions,  names  of  materials, 
dates,  etc.,  should  be  made  free  hand. 

There  are  many  styles  of  letters  used  by  draftsmen.  For 
titles,  large  Roman  capitals  are  frequently  used,  although  Gothic 
and  block  letters  also  look  well  and  are  much  easier  to  make. 

ABCDEFGHIJ 
KLMNOPQR 
STUVWXYZ 

1234567890 


Almost  any  neat  letter  free  from  ornamentation  is  acceptable  in  the 
regular  practice  of  drafting.  Fig.  32  shows  the  alphabet  oi 
vertical  Gothic  capitals.  These  letters  are  neat,  plain  and  easily 
made.  The  inclined  or  italicized  Gothic  type  is  shown  in  Fig.  33. 
This  style  is  also  easy  to  construct,  and  possesses  the  advantage 
that  a  slight  difference  in  inclination  is  not  apparent.  If  the  ver- 
tical lines  of  the  vertical  letters  incline  slightly  the  inaccuracy  is 
very  noticeable. 

The  curves  of  the  inclined  Gothic  letters  such  as  those  in  the 
B,  C,  Gr,  e7,  etc.,  are  somewhat  difficult  to  make  free  hand, 
especially  if  the  letters  are  about  one-half  inch  high.  In  the 
alphabet  shown  in  Fig.  34,  the  letters  are  made  almost  wholly  of 


MECHANICAL  DRAWING. 


straight  lines,  the  corners  only  being  curved.  These  letters  are 
very  easy  to  make  and  are  clear  cut. 

The  first  few  plates  of  this  work  will  require  no  titles;  the 
only  lettering  being  the  student's  name,  together  with  the  date 
and  plate  number.  Later,  the  student  will  take  up  the  subject  of 

ABCDETGH/J 
KLMNOPQFt 
STUVWXYZ 

Fig.  3.3. 

lettering  again  in  order  to  letter  titles  and  headings  for  drawings 
showing  the  details  of  machines.  For  the  present,  however,  in- 
clined Gothic  capitals  will  be  used. 

To  make  the  inclined  Gothic  letters,  first  draw  two  parallel 
lines  having  the  distance  between  them  equal  to  the  desired  height 
of  the  letters.  If  two  sizes  of  letters  are  to  be  used,  the  smaller 
should  be  about  two-thirds  as  high  as  the  larger.  For  the  letters 

A  BCDETGH/JKLM 

NOPQFISTUVWXYZ 

/23456789O 

Fig.  34. 

to  be  used  on  the  first  plates,  draw  two  parallel  lines  ^  inch  apart. 
This  is  the  height  for  the  letters  of  the  date,  name,  also  the  plat^ 
number,  and  should  be  used  on  all  plates  throughout  this  ^-ork, 
unless  other  directions  are  given. 

In  constructing  the  letters,  they  should  extend  fully  to  these 
lines,  both  at  the  top  and  bottom.  They  should  not  fall  short  of 


231 


24  MECHANICAL    DRAWING. 

the  guide  lines  nor  extend  beyond  them.  As  these  letters  are 
inclined  they  will  look  better  if  the  inclination  is  the  same  for  all. 
As  an  aid  to  the  beginner,  he  can  draw  light  pencil  lines,  about  \ 
inch  apart,  forming  the  proper  angle  with  the  parallel  lines  already 
drawn.  The  inclination  is  often  made  about  70  degrees ;  but  as  a 
60-degree  triangle  is  at  hand,  it  may  be  used.  To  draw  these 
lines  place  the  60-degree  triangle  on  the  T-square  as  shown  in 
Fig.  36.  In  making  these  letters  the  60-degree  lines  will  be 
found  a  great  aid  as  a  large  proportion  of  the  back  or  side  lines 
have  this  inclination. 

Capital  letters  such  as  E,  jP,  P,  T,  Z,  etc.,  should  have  the 
top  lines  coincide  with  the  upper  horizontal  guide  line.  The 
bottom  lines  of  such  letters  as  Z>,  E,  L,  Z,  etc.,  should  coincide 
with  the  lower  horizontal  guide  line.  If  these  lines  do  not  coin- 
cide with  the  guide  lines  the  words  will  look  uneven.  Letters, 
of  which  (7,  6r,  0,  and  Q,  are  types,  can  be  formed  of  curved  lines 
or  of  straight  lines.  If  made  of  curved  lines,  they  should  have  a 
little  greater  height  than  the  guide  lines  to  prevent  their  appear- 
ing  smaller  than  the  other  letters.  In  this  work  they  can  be 
made  of  straight  lines  with  rounded  corners  as  they  are  easily 
constructed  and  the  student  can  make  all  letters  of  the  same 
height. 

To  construct  the  letter  A,  draw  a  line  at  an  angle  of  60 
degrees  to  the  horizontal  and  use  it  as  a  center  line.  Then  from 
the  intersection  of  this  line  and  the  upper  horizontal  line  drop 
a  vertical  line  to  the  lower  guide  line.  Draw  another  line  from 
the  vertex  meeting  the  lower  guide  line  at  the  same  distance  from 
the  center  line.  The  cross  line  of  the  A  should  be  a  little  below 
the  center.  The  F"is  an  inverted  A  without  the  cross  line.  For 
the  letter  M,  the  side  lines  should  be  parallel  and  about  the  same 
distance  apart  as  are  the  horizontal  lines.  The  side  lines  of  the 
IF  are  not  parallel  but  are  farther  apart  at  the  top.  The  «7is  not 
quite  as  wide  as  such  letters  as  H,  E^  N,  R,  etc.  To  make  a  Y. 
draw  the  center  line  60  degrees  to  the  horizontal ;  the  diverg- 
ing lines  are  similar  to  those  of  the  V  but  are  shorter  and  form  a 
larger  angle.  The  diverging  lines  should  meet  the  center  line  a 
little  below  the  middle. 

Tlie  lower-case  letters  are  shown  in  Fig.  35.     In  such  letters 


232 


MECHANICAL    DRAWING.  26 

as  7/1,  n,  r,  etc.,  make  the  corners  sharp  and  not  rounding.  The 
letters  a,  b,  c,  e,  g,  o,  p,  q,  should  be  full  and  rounding.  The 
figures  (see  Fig.  32)  are  made  as  in  writing  —  except  the  4,  tf,  8 
and  9. 

The  Roman  numerals  are  made  of  straight  lines  as  they 
are  largely  made  up  of  I,  F"and  X. 

At  first  the  copy  should  be  followed  closely  and  the  letters 
flrawn  in  pencil.  For  a  time,  the  inclined  guide  lines  may  be  used. 

obcctefgh/jk/mn 


Fig.  35. 

but  after  the  proper  inclination  becomes  firmly  fixed  in  mind 
they  should  be  abandoned.  The  horizontal  lines  are  used  at  all 
times  by  most  draftsmen.  After  the  student  has  had  consider- 
able practice,  he  can  construct  the  letters  in  ink  without  first  using 
the  pencil.  Later  in  the  work,  when  the  student  has  become  pro- 
ficient in  the  simple  inclined  Gothic  capitals,  the  vertical,  block 
and  Roman  alphabets  should  be  studied. 

PLATES, 

To  lay  out  a  sheet  of  paper  for  the  plates  of  this  work,  the 
sheet,  A  B  G  F,  (Fig.  36)  is  placed  on  the  drawing  board  2  or  3 
inches  from  the  left-hand  edge  which  is  called  the  working  edge. 
If  placed  near  the  left-hand  edge,  the  T-square  and  triangles  can 
be  used  with  greater  firmness  and  the  horizontal  lines  drawn  with 
the  T-square  will  be  more  accurate.  In  placing  the  paper  on  the 
board,  always  true  it  up  according  to  the  long  edge  of  the  sheet. 
First  fasten  the  paper  to  the  board  with  thumb  tacks,  using  at 
least  4  —  one  at  each  corner.  If  the  paper  has  a  tendency  to  curl 
it  is  better  to  use  6  or  8  tacks,  placing  them  as  shown  in  Fig.  36. 
Thumb  tacks  are  commonly  used  ;  but  many  draftsmen  prefer 
one-ounce  tacks  as  they  offer  less  obstruction  to  the  T-square  and 
triangles. 

After  the  paper  is  fastened  in  position,  find  the  center  of  the 


233 


MECHANICAL     DRAWING. 


Fig. 


234 


MECHANICAL    DRAWING.  27 

sheet  by  placing  the  T-square  so  that  its  upper  edge  coincides  with 
the  diagonal  corners  A  and  G  and  then  with  the  corners  F  and 
B,  drawing  short  pencil  lines  intersecting  at  C.  Now  place  the 
T-square  so  that  its  upper  edge  coincides  with  the  point  C  and 
draw  the  dot  and  dash  line  D  E.  With  the  T-square  and  one 
of  the  triangles  (shown  dotted)  in  the  position  shown  in  Fig.  36, 
draw  the  dot  and  dash  line  H  C  K.  In  case  the  drawing  board 
is  large  enough,  the  line  C  H  can  be  drawn  by  moving  the  T- 
square  until  it  is  entirely  off  the  drawing.  If  the  board  is  small, 
produce  (extend)  the  line  K  C  to  H  by  means  of  the  T-square 
or  edge  of  a  triangle.  In  this  work  always  move  the  pencil  from 
the  left  to  the  right  or  from  the  bottom  upward ;  never  (except 
for  some  particular  purpose)  in  the  opposite  direction. 

After  the  center  lines  are  drawn  measure  off  5  inches  above 
and  below  the  point  C  on  the  line  JI  G  K.  These  points  L 
and  M  may  be  indicated  by  a  light  pencil  mark  or  by  a  slight 
puncture  of  one  of  the  points  of  the  dividers.  Now  place  the  T-- 
square against  the  left-hand  edge  of  the  board  and  draw  horizontal 
pencil  lines  through  L  and  M. 

Measure  off  7  inches  to  the  left  and  right  of  C  on  the  center 
line  DOE  and  draw  pencil  lines  through  these  points  (N  and 
P)  perpendicular  to  D  E.  We  now  have  a  rectangle  10  inches 
by  14  inches,  in  which  all  the  exercises  and  figures  are  to  be 
drawn.  The  lettering  of  the  student's  name  and  address,  date, 
and  plate  number  are  to  be  placed  outside  of  this  rectangle  in  the 
^-inch  margin.  In  all  cases  lay  out  the  plates  in  this  manner  and 
keep  the  center  lines  D  E  and  K  H  as  a  basis  for  the  various 
figures.  The  border  line  is  to  be  inked  with  a  heavy  line  when 
the  drawing  is  inked. 

Pencilling.  Inlaying  out  plates,  all  work,  is  first  done  in  pen 
cil  and  afterward  inked  or  traced  on  tracing  cloth.  The  first  few 
plates  of  this  course  are  to  be  done  in  pencil  and  then  inked  ;  later 
the  subject  of  tracing  and  the  process  of  making  blue  prints  will 
be  taken  up.  Every  beginner  should  practice  with  his  instruments 
until  he  can  use  them  with  accuracy  and  skill,  and  until  he  under- 
stands thoroughly  what  instrument  should  be  used  for  making  a 
given  line.  To  aid  the  beginner  in  this  work,  the  first  three  plates 
of  this  course  are  designed  to  give  the  student  practice :  they  do 


MECHANICAL    DRAWING. 


not  involve  any  problems  and  none  of  the  work  is  difficult.  The 
student  is  strongly  advised  to  draw  these  plates  two  or  three 
times  before  making  the  one  to  be  sent  to  us  for  correction.  Dili- 
gent practice  is  necessary  at  first;  especially  on  PLATE  I as  it 
involves  an  exercise  in  lettering. 

PLATE  I. 

Pencilling.  To  draw  PLATE  7,  take  a  sheet  of  drawing 
paper  at  least  11  inches  by  15  inches  and  fasten  it  to  the  drawing 
board  as  already  explained.  Find  the  center  of  the  sheet  and  draw 
fine  pencil  lines  to  represent  the  lines  D  E  and  H  K  of  Fig.  36. 
Also  draw  the  border  lines  L,  M,  N  and  P. 

Now  measure  |  inch  above  and  below  the  horizontal  center  line 
and,  with  the  T-square,  draw  lines  through  these  points.  These 
lines  will  form  the  lower  lines  D  C  of  Figs.  1  and  2  and  the  top  lines 
A  B  of  Figs.  3  and  4-  Measure  |  inch  to  the  right  and  left  of  the 
vertical  center  line ;  and  through  these  points,  draw  lines  parallel 
to  the  center  line.  These  lines  should  be  drawn  by  placing  the 
triangle  on  the  T-square  as  shown  in  Fig.  36.  The  lines  thus 
drawn,  form  the  sides  B  C  of  Figs.  1  and  3  and  the  sides  A  D  of 
Figs.  2  and  4-  Next  draw  the  line  A  B  A  B  with  the  T-square, 
4 1  inches  above  the  horizontal  center  line.  This  line  forms  the 
top  lines  of  Figs.  1  and  2.  Now  draw  the  line  D  C  D  C  4|  inches 
below  the  horizontal  center  line.  The  rectangles  of  the  four 
figures  are  completed  by  drawing  vertical  lines  6|  inches  from  the 
vertical  center  line.  We  now  have  four  rectangles  each  6 \  inches 
long  and  4|  inches  wide. 

Fig.  1  is  an  exercise  with  the  line  pen  and  T-square.  Divide 
the  line  AD  into  divisions  each  J  inch  long,  making  a  fine  pencil 
point  or  slight  puncture  at  each  division  such  as  E,  F,  G,  H,  I,  etc. 
Now  place  the  T-square  with  the  head  at  the  left-hand  edge  of  the 
drawing  board  and  through  these  points  draw  light  pencil  lines 
extending  to  the  line  B  C.  In  drawing  these  lines  be  careful  to 
have  the  pencil  point  pass  exactly  through  the  division  marks  so 
that  the  lines  will  be  the  same  distance  apart.  Start  each  line  iu 
the  line  A  D  and  do  not  fall  short  of  the  line  B  C  or  run  over  it. 
Accuracy  and  neatness  in  pencilling  insure  an  accurate  drawing. 
Some  beginners  think  that  they  can  correct  inaccuracies  while 


_<. _  Q    I    <  n 


MECHANICAL     DRAWING. 


inking;  but  experience  soon  teaches  them  that  they  cannot  do  so. 
Fig.  2  is  an  exercise  with  the  line  pen,  T-square  and  triangle. 
First  divide  the  lower  line  D  C  of  the  rectangle  into  divisions  each 
|  inch  long  and  mark  the  points  E,  F,  G,  H,  I,  J,  K,  etc.,  as  in 
Fig.  1.  Place  the  T-square  with  the  head  at  the  left-hand  edge  of 
the  drawing  board  and  the  upper  edge  in  about  the  position  shown 
in  Fig.  36.  Place  either  triangle  with  one  edge  on  the  upper  edge 
of  the  T-square  and  the  90-degree  angle  at  the  left.  Now  draw 
fine  pencil  lines  from  the  line  D  C  to  the  line  A  B  passing  through 
the  points  E,  F,  G,  H,  I,  J,  K,  etc.  To  do  this  keep  the  T-square 


1 

f 

r 

" 

//•/// 

5!Z52Za! 

/  /-ft.      SLJBiJ 

^ 

(-?'{-?  /-  A  / 

A  N/CA  Z_ 

f-  jcf-  /  ?  t  y 

/7/tc7/i"  Sf-* 

ABODE 

/  ^  ^4- 

/  //  ///  /i/ 

z 

X 

B 


rigid  and  slide  the  triangle  toward  the  right,  being  careful  to  have 
the  edge  coincide  with  the  division  marks  in  succession. 

Fig.  3  is  an  exercise  with  the  line  pen,  T-square  and  45-degree 
triangle.  First  lay  off  the  distances  A  E,  E  F,  F  G,  G  H,  H  I,  IJ, 
J  K,  etc.,  each  \  inch  long.  Then  lay  off  the  distances  B  L,  L  M, 
M  N,  N  O,  O  P,  P  Q,  Q  R,  etc.,  also  \  inch  long.  Place  the  T- 
square  so  that  the  upper  edge  will  be  below  the  line  D  C  of  Fig.  3. 
With  the  45-degree  triangle  draw  lines  from  A  D  and  D  C  to 
the  points  E,  F,  G,  H,  I,  J,  K,  etc.,  as  far  as  the  point  B.  Now 
draw  lines  from  D  C  to  the  points  L,  M,  N,  O,  P,  Q,  R,  etc.,  as 


239 


MECHANICAL    DRAWING. 


far  as  the  point  C.  In  drawing  these  lines  move  the  pencil  away 
from  the  body,  that  is,  from  A  D  to  A  B  and  from  D  C  to  B  C. 

Fig.  4  is  an  exercise  in  free-hand  lettering.  The  finished 
exercise,  with  all  guide  lines  erased,  should  have  the  appearance 
shown  in  Fig.  4  of  PL  A  TE  I.  The  guide  lines  are  drawn  as  shown 
in  Fig,  87.  First  draw  the  center  line  E  F  and  light  pencil  lines 
Y  Z  and  T  X,  f  inch  from  the  border  lines.  Now,  with  the  T- 
square,  draw  the  line  G,  \  inch  from  the  top  line  and  the  line  H, 
^  inch  below  G.  The  word  "LETTERING-"  is  to  be  placed 
between  these  two  lines.  Draw  the  line  I,  fa  inch  below  H. 
The  lines  I,  J,  etc.,  to  K  are  all  fa  inch  apart. 

We  now  practice  the  lower-case  letters.  Draw  the  line  L,  -^ 
inch  below  K  and  a  light  line  |  inch  above  L  to  limit  the 
height  of  the  small  letters.  The  space  between  L  and  M  is  fa 
inch.  The  lines  M  and  N  are  drawn  in  the  same  manner  as  K  and 
L.  The  space  between  N  and  O  should  be  ^  inch.  The  line  P  is 
drawn  fa  inch  below  O.  Q  is  also  fa  inch  below  P.  The  lines 
Q  and  R  are  drawn  Jg  inch  apart  as  are  M  and  N.  The  remainder 
of  the  lines  S,  U,  V  and  W  are  drawn  fa  inch  apart. 

The  center  line  is  a  great  aid  in  centering  the  word 
« LETTERING"  the  alphabets,  numerals,  etc.  The  words 
"THE"  and  "Proficiency"  should  be  indented  about  f 
inch  as  they  are  the  first  words  of  paragraphs.  To  draw  the 
guide  lines,  mark  off  distances  of  £  inch  on  any  line  such  as  J  and 
with  the  60-degree  triangle  draw  light  pencil  lines  cutting  the 
parallel  lines.  The  letters  should  be  sketched  in  pencil,  the  ordin- 
ary letters  such  as  E,  F,  H,  N,  R,  etc.  being  made  of  a  width 
equal  to  about  f  the  height.  Letters  like  A,  M  and  W  are  wider. 
The  space  between  the  letters  depends  upon  the  draftsman's 
taste  but  the  beginner  should  remember  that  letters  next  to  an 
A  or  an  L  should  be  placed  near  them  and  that  greater  space 
should  be  left  on  each  side  of  an  I  or  between  letters  whose  sides  are 
parallel;  for  instance  there  should  be  more  space  between  an  N  and 
E  than  between  an  E  and  H.  On  account  of  the  space  above  the 
lower  line  of  the  L,  a  letter  following  an  L  should  be  close  to  it 
If  a  T  follows  a  T  or  the  letter  L  follows  an  L  they  should  lie 
placed  near  together.  In  all  lettering  the  letters  should  be  placed 
BO  that  the  general  effect  is  pleasing.  After  the  four  figures  are 


240 


MECHANICAL    DRAWING.  31 

completed,  the  lettering  for  name,  address  and  date  should  be 
pencilled.  With  the  T-square  draw  a  pencil  line  ^  inch  above 
the  top  border  line  at  the  right-hand  end.  This  line  should  be 
about  3  inches  long.  At  a  distance  of  fa  inch  above  this  line  draw 
another  line  of  about  the  same  length.  These  are  the  guide  lines 
for  the  word  PLATE  I.  The  letters  should  be  pencilled  free 
hand  and  the  student  may  use  the  60-degree  guide  lines  if  he 
desires. 

The  guide  lines  of  the  date,  name  and  address  are  similarly 
drawn  in  the  lower  margin.  The  date  of  completing  the  drawing 
should  be  placed  under  Fig.  3  and  the  name  and  address  at  the 
right  under  Fig.  4-  The  street  address  is  unnecessary.  It  is  a 
good  plan  to  draw  lines  fa  inch  apart  on  a  separate  sheet  of  paper 
and  pencil  the  letters  in  order  to  know  just  how  much  space  each 
word  will  require.  The  insertion  of  the  words  «'  Fig.  jf,"  "  Fig. 
2"  etc.,  is  optional  with  the  student.  He  may  leave  them  out  if  he 
desires  ;  but  we  would  advise  him  to  do  this  extra  lettering  for  the 
practice  and  for  convenience  in  reference.  First  draw  with  the 
T-square  two  parallel  line  fa  inch  apart  under  each  exercise ;  the 
lower  line  being  ^  inch  above  the  horizontal  center  line  or  above 
the  lower  border  line. 

Inking.  After  all  of  the  pencilling  of  PLATE  I  has  been 
completed  the  exercises  should  be  inked.  The  pen  should  first  be 
examined  to  make  sure  that  the  nibs  are  clean,  of  the  same  length 
and  come  together  evenly.  To  fill  the  pen  with  ink  use  an  ordi- 
nary steel  pen  or  the  quill  in  the  bottle,  if  Higgin's  Ink  is  used. 
Dip  the  quill  or  pen  into  the  bottle  and  then  inside  between  the 
nibs  of  the  line  pen.  The  ink  will  readily  flow  from  the  quill  into 
the  space  between  the  nibs  as  soon  as  it  is  brought  in  contact.  Do 
not  fill  the  pen  too  full,  if  the  ink  fills  about  \  the  distance  to  the 
adjusting  screw  it  usually  will  be  sufficient.  If  the  filling  has  been 
carefully  done  it  will  not  be  necessary  to  wipe  the  outsides  of  the 
blades.  However,  any  ink  on  the  outside  should  be  wiped  off 
with  a  soft  cloth  or  a  piece  of  chamois. 

The  pen  should  now  be  tried  on  a  separate  piece  of  paper  in 
order  that  the  width  of  the  line  may  be  adjusted.  In  the  first 
work  where  no  shading  is  done,  a  firm  distinct  line  should  be  used. 
The  beginner  should  avoid  the  extremes ;  a  very  light  line  makes 


241 


MECHANICAL    DRAWING. 


the  drawing  have  a  weak,  indistinct  appearance,  and  very  heavy 
lines  detract  from  the  artistic  appearance  and  make  the  drawing 
appear  heavy. 

In  case  the  ink  does  not  flow  freely,  wet  the  finger  and  touch 
it  to  the  end  of  the  pen.  If  it  then  fails  to  flow,  draw  a  slip  of 
thin  paper  between  the  nibs  (thus  removing  the  dried  ink)  or 
clean  thoroughly  and  fill.  Never  lay  the  pen  aside  without 
cleaning. 

In  ruling  with  the  line  pen  it  should  be  held  firmly  in  the 
right  hand  almost  perpendicular  to  the  paper.  If  grasped  too 
firmly  the  width  of  the  line  may  be  varied  and  the  draftsman 
soon  becomes  fatigued.  The  pen  is  usually  held  so  that  the 
adjusting  screw  is  away  from  the  T-square,  triangles,  etc.  Many 
draftsmen  incline  the  pen  slightly  in  the  direction  in  which  it  is 
moving. 

To  ink  Fig.  1,  place  the  T-square  with  the  head  at  the  work- 
ing edge  as  in  pencilling.  First  ink  all  of  the  horizontal  lines 
moving  the  T-square  from  A  to  D.  In  drawing  these  lines  con- 
siderable  care  is  necessary ;  both  nibs  should  touch  the  paper  and 
the  pressure  should  be  uniform.  Have  sufficient  ink  in  the  pen 
to  finish  the  line  as  it  is  difficult  for  a  beginner  to  stop  in  the 
middle  of  the  line  and  after  refilling  the  pen  make  a  smooth  con- 
tinuous line.  While  inking  the  lines  A,  E,  F,  G,  H,  I,  etc.,  greater 
care  should  be  taken  in  starting  and  stopping  than  while  pencil- 
ling. Each  line  should  start  exactly  in  the  pencil  line  A  D  and 
stop  in  the  line  B  C.  The  lines  A  D  and  B  C  are  inked,  using 
the  triangle  and  T-square. 

Fig.  2  is  inked  in  the  same  manner  as  it  was  pencilled;  the 
lines  being  drawn,  sliding  the  triangle  along  the  T-square  in  the 
successive  positions. 

In  inking  Fig.  3,  the  same  care  is  necessary  as  with  the  pre- 
ceding, and  after  the  oblique  lines  are  inked  the  border  lines  are 
finished.  In  Fig.  4  the  border  lines  should  be  inked  in  first 
and  then  the  border  lines  of  the  plate.  The  border  lines  should 
be  quite  heavy  as  they  give  the  plate  a  better  appearance.  The 
intersections  should  be  accurate,  as  any  running  over  necessitates 
erasing. 

The  line  pen  may  now  be  cleaned  and  laid  aside.     It  can  be 


242 


r* 


MECHANICAL    DRAWING. 


cleaned  by  drawing  a  strip  of  blotting  paper  between  the  nibs  or 
by  means  of  a  piece  of  cloth  or  chamois.  The  lettering  should  be 
done  free-hand  using  a  steel  pen.  If  the  pen  is  very  fine,  accu- 
rate work  may  be  done  but  the  pen  is  likely  to  catch  in  the  paper, 
especially  if  the  paper  is  rough.  A  coarser  pen  will  make  broader 
lines  but  is  on  the  whole  preferable.  Gillott's  404  is  as  fine  a 
pen  as  should  be  used.  After  inking  Fig.  4,  the  plate  number, 
date  and  name  should  be  inked,  also  free-hand.  After  ink- 
ing the  words  "  Fig.  1"  "  Fig.  2,"  etc.,  all  pencil  lines  should 
ba  erased.  In  the  finished  drawing  there  should  be  no  center 
lines,  construction  lines  or  letters  other  than  those  in  the 
name,  date,  etc. 

The  sheet  should  be  cut  to  a  size  of  u  inches  by  15  inches, 
the  dash  line  outside  the  border  line  of  PLATE  I. indicating  the 
edge. 

PLATE  II. 

Pencilling.  The  drawing  paper  used  for  PLATE  //should 
be  laid  out  as  described  with  PLATE  I,  that  is,  the  border  lines, 
center  line  and  rectangles  for  Figs.  1  and  2.  To  lay  out  Figs.  3, 
4  and  5  proceed  as  follows  :  Draw  a  line  with  the  T-square 
parallel  to  the  horizontal  center  line  and  |  inch  below  it.  Also 
draw  another  similar  line  4f  below  the  center  line.  The  two  lines 

o 

will  form  the  top  and  bottom  of  Figs.  3,  £  and  5.  Now  measure 
off  2^  inches  on  either  side  of  the  center  on  the  horizontal  center 
line  and  call  the  points  Y  and  Z.  On  either  side  of  Y  and  Z  and 
at  a  distance  of  J  inch  draw  vertical  parallel  lines.  Now  draw  a 
vertical  line  A  D,  4^  inches  from  the  line  Y  and  a  vertical  line 
B  C  4|  inches  from  the  line  Z.  We  now  have  three  rectangles 
each  4  inches  broad  and  4|  inches  high.  Figs.  1  and  2  are  pen- 
cilled in  exactly  the  same  way  as  was  Fig.  1  of  PLATE  /,  that 
is,  horizontal  lines  are  drawn  |  inch  apart. 

Fig.  3  is  an  exercise  to  show  the  use  of  a  60-degree  triangle 
with  a  T-square.  Lay  off  the  distances  A  E,  E  F,  F  G,  G  H,  etc. 
to  B  each  J  inch.  With  the  60  degree  triangle  resting  on  the 
upper  edge  of  the  T-square,  draw  lines  through  these  points,  E,  F, 
G,  H,  I,  J,  etc.,  forming  an  angle  of  30  degrees  with  the  hori- 
zontal. The  last  line  drawn  will  be  A  L.  In  drawing  these  lines 
move  the  poncil  from  A  B  to  B  C.  Now  find  the  distance 


245 


.34  MECHANICAL    DRAWING. 

between  the  lines  on  the  vertical  B  L  and  mark  off  these  distances 
on  the  line  B  C  commencing  at  L.  Continue  the  lines  from  A  L 
to  N  C.  Commencing  at  N  mark  off  distances  on  A  D  equal 
to  those  on  B  C  and  finish  drawing  the  oblique  lines. 

Fig.  4  is  an  exercise  for  intersection.  Lay  off  distances  of 
|  inch  on  A  B  and  A  D.  With  the  T-square  draw  fine  pencil 
lines  through  the  points  E,  F,  G,  H,  I,  etc.,  and  with  the  T-square 
and  triangle  draw  vertical  lines  through  the  points  L,  M,  N,  O,  P, 
etc.  In  drawing  this  figure  draw  every  line  exactly  through  the 
points  indicated  as  the  symmetrical  appearance  of  the  small 
squares  can  be  attained  only  by  accurate  pencilling. 

The  oblique  lines  in  Fig.  5  form  an  angle  of  60  degrees  with 
the  horizontal.  As  in  Figs.  3  and  4  mark  off  the  line  A  B  in 
divisions  of  ^  inch  and  draw  with  the  T-square  and  60-degree 
triangle  the  oblique  lines  through  these  points  of  division  moving 
the  pencil  from  A  B  to  B  C.  The  last  line  thus  drawn  will  be 
A  L.  Now  mark  off  distances  of  ^  inch  on  C  D  beginning  at  L. 
The  lines  may  now  be  finished. 

Inking.  Fig.  1  is  designed  to  give  the  beginner  practice  in 
drawing  lines  of  varying  widths.  The  line  E  is  first  drawn.  This 
line  should  be  rather  fine  but  should  be  clear  and  distinct.  The 
line  F  should  be  a  little  wider  than  E ;  the  greater  width  being 
obtained  by  turning  the  adjusting  screw  from  one-quarter  to  one- 
half  a  turn.  The  lines  G,  H,  I,  etc.,  are  drawn ;  each  successive 
Jine  having  greater  width.  M  and  N  should  be  the  same  and 
quite  heavy.  From  N  to  D  the  lines  should  decrease  in  width. 
To  complete  the  inking  of  Fig.  1,  draw  the  border  lines.  These 
lines  should  have  about  the  same  width  as  those  in  PLATE  I. 

In  Fig.  2  the  first  four  lines  should  be  dotted.  The  dots  should 
be  uniform  in  length  (about  -^  inch)  and  the  spaces  also  uniform 
(about  -gig-  inch).  The  next  four  lines  are  dash  lines  similar  to 
those  used  for  dimensions.  These  lines  should  be  drawn  with 
dashes  about  |  inch  long  and  the  lines  should  be  fine,  yet  distinct. 

The  following  four  lines  arc  called  dot  and  dash  lines.  The 
dashes  are  about  |  inch  long  and  a  dot  between  as  shown.  In 
the  regular  practice  of  drafting  the  length  of  the  dashes  depends 
upon  the  size  of  the  drawing  —  ^  inch  to  1  inch  being  common. 
The  last  tour  lines  are  similar,  two  dots  being  used  between  the 


248 


MECHANICAL    DRAWING.  35 

dashes.  After  completing  the  dot  and  dash  lines,  draw  the  border 
lines  of  the  rectangle  as  before. 

In  inking  Fig.  3,  the  pencil  lines  are  followed.  Great  care 
should  be  exercised  in  starting  and  stopping.  The  lines  should 
begin  in  the  border  lines  and  the  end  should  not  run  over. 

The  lines  of  Fig.  4-  must  be  drawn  carefully,  as  there  are  so 
many  intersections.  The  lines  in  this  figure  should  be  lighter  than 
the  border  lines.  If  every  line  does  not  coincide  with  the  points 
of  division  L,  M,  N,  O,  P,  etc.,  some  will  appear  farther  apart 
than  others. 

Fig.  5  is  similar  to  Fig.  3,  the  only  difference  being  in  the 
angle  which  the  oblique  lines  make  with  the  horizontal. 

After  completing  the  five  figures  draw  the  border  lines  of  the 
plate  and  then  letter  the  plate  number,  date  and  name,  and  the 
figure  numbers,  as  in  PLATE  I.  The  plate  should  then  be 
cut  to  the  required  size,  11  inches  by  15  inches. 

PLATE  III. 

Pencilling.  The  horizontal  and  vertical  center  lines  and  the 
border  lines  for  PLATE  III  are  laid  out  in  the  same  manner  as 
were  those  of  PLATE  II.  To  draw  the  squares  for  the  six  figures, 
proceed  as  follows  : 

Measure  off  two  inches  on  either  side  of  the  vertical  center 
line  and  draw  light  pencil  lines  through  these  points  parallel  to 
the  vertical  center  line.  These  lines  will  form  the  sides  A  D  and 
B  C  of  Figs.  2  and  5.  Parallel  to  these  lines  and  at  a  distance  of 
I  inch  draw  similar  lines  to  form  the  sides  B  C  of  Figs.  1  and  If 
and  A  D  of  Figs.  3  and  6.  The  vertical  sides  A  D  of  Figs.  1  and 
4  and  B  C  of  Figs.  3  and  6  are  formed  by  drawing  lines  perpen- 
dicular to  the  horizontal  center  line  at  a  distance  of  6^  inches  from 
the  center. 

The  horizontal  sides  D  C  of  Figs.  1,  2  and  3  are  drawn  with 
the  T-square  A  inch  above  the  horizontal  center  line.  To  draw  the 
top  lines  of  these  figures,  draw  (with  the  T-square)  a  line  4J  inches 
above  the  horizontal  center  line.  The  top  lines  of  Figs.  4,  5  and 
6  are  drawn  £  inch  below  the  horizontal  center  line.  The  squares 
are  completed  by  drawing  the  lower  lines  D  C,  4|  inches  below 
the  horizontal  center  line.  The  figures  of  PLATES  I  and  U 


247 


MECHANICAL    DRAWING. 


were  constructed  in  rectangles;  the  exercises  of  PLATE  JZ/are, 
however,  drawn  in  squares,  having  the  sides  4  inches  long. 

In  drawing  Fig.  1,  first  divide  A  D  and  A  B  (or  D  C  )  into 
4  equal  parts.  As  these  lines  are  four  inches  long,  each  length  will 
be  1  inch.  Now  draw  horizontal  lines  through  E,  F  and  G  and 
vertical  lines  through  L,  M  and  N.  These  lines  are  shown  dotted 
in  Fig.  1.  Connect  A  and  B  with  the  intersection  of  lines  E 
and  M,  and  A  and  D  with  the  intersection  of  lines  F  and  L. 
Similarly  drav  D  J,  J  C,  I  B  and  I  C.  Also  connect  the  points  P, 
O,  I  and  J  forming  a  square.  The  four  diamond  shaped  areas 
are  formed  by  drawing  lines  from  the  middle  points  of  A  D,  A  B, 
B  C  and  D  C  to  the  middle  points  of  lines  A  P,  A  O,  O  B,  I  B 
etc.,  as  shown  in  Fi<j.  1. 

Fig.  %  is  an  exercise  of  straight  lines.  Divic'e  A  D  and  A  B 
into  four  equal  parts  and  draw  horizontal  and  vertical  lines  as  in 
Fig.  1.  Now  divide  these  dimensions,  A  L,  M  N,  etc.  and  E  F, 
G  B  etc.  into  four  equal  parts  (each  ^  inch).  Draw  light 
pencil  lines  with  the  T-square  and  triangle  as  s.hown  in  Fig.  2. 

In  Fig.  3,  divide  A  B  and  A  D  into  eight,  parts,  each  length 
being  ^  inch.  Through  the  points  H,  I,  J,  K,  L,  M  and  N  draw 
vertical  lines  with  the  triangle.  Through  O,  P,  Q,  R,  S,  T  and  U 
draw  horizontal  lines  with  the  T-square.  Now  draw  lines  con- 
necting O  and  H,  P  and  I,  Q  and  J,  etc.  These  lines  can  be 
drawn  with  the  45-degree  triangle,  as  thoy  form  an  angle  of  45 
degrees  with  the  horizontal.  Starting  at  N  draw  lines  from  A  B 
to  B  C  at  an  angle  of  45  degrees.  Also  draw  lines  from  A  D  to 
D  C  through  the  points  O,  P,  Q,  R,  ete.,  forming  angles  of  45 
degrees  with  D  C. 

Fig.  4  is  drawn  with  the  compasses.  First  draw  the  diagonals 
A  C  and  D  B.  With  the  T^quare  <iraw  the  line  E  H.  Now 
mark  off  on  E  H  distances  of  |  inch.  With  the  compasses  set  so 
that  the  point  of  the  lead  is  '2  inches  from  the  needle  point,  de- 
scribe the  circle  passing  thro  igh  E  With  H  as  a  center  draw 
the  arcs  F  G  and  I  J  having  a  radius  of  1|  inches.  In  drawing 
these  arcs  be  careful  not  to  go  beyond  the  diagonals,  but  stop  at 
the  points  F  and  G  and  I  and  J.  Again  with  H  as  the  center 
and  a  radius  of  1|  inches  draw  a  circle.  The  arcs  K  L  and  M  N 
are  drawn  in  the  same  manner  as  were  arcs  P  G  and  I  J ;  the 


248 


O      DL     Q     ft     CO      t-     D 


MECHANICAL    DRAWING.  87 

radius  being  1J  inches.     Now  draw  circles,  with  H  as  the  center, 
of  1,  |,  i  and  I  incli  radius,  passing  through  the  points  P,  T,  etc. 

Fig.  5  is  an  exercise  with  the  line  pen  and  compasses.  First 
draw  the  diagonals  A  C  and  D  B,  the  horizontal  line  L  M  and  the 
vertical  line  E  F  passing  through  the  center  Q.  Mark  off  dis- 
tances of  I  inch  on  L  M  and  E  F  and  draw  the  lines  N  N'  O  O 
and  N  R,  O  S,  etc.,  through  these  points,  forming  the  squares 
N  R  R'  N ',  O  S  S'  O',  etc.  With  the  bow  pencil  adjusted  so 
that  the  distance  between  the  pencil  point  and  the  needle  point  is 
I  inch  draw  the  arcs  having  centers  at  the  corners  of  the  squares. 
The  arc  whose  center  is  N  will  be  tangent  to  the  lines  A  L  and 
A  E  and  the  arc  whose  center  is  O  will  be  tangent  to  N  N'  and 
N  R.  Since  P  T,  T  T',  T'  P'  and  P'  P  are  each  1  inch  long  and 
form  the  square,  the  arcs  drawn  with  Q  as  a  center  will  form  a 
circle. 

To  draw  Fig.  tf,  first  draw  the  center  lines  E  F  and  L  M. 
Now  find  the  centers  of  the  small  squares  ALIE,  LBFI  etc. 
Through  the  center  I  draw  the  construction  lines  HIT  and 
RIP  forming  angles  of  30  degrees  with  the  horizontal.  Now 
adjust  the  compasses  to  draw  circles  having  a  radius  of  one  inch. 
With  I  as  a  center,  draw  the  circle  H  P  T  R.  With  the  same 
radius  ( one  inch  )  draw  the  arcs  with  centers  at  A,  B,  C  and 
D.  Also  draw  the  semi-circles  with  centers  at  L,  F,  M  and  E. 
Now  draw  the  arcs  as  shown  having  centers  at  the  centers  of  the 
small  squares  A  L  I  E,  L  B  F  I,  etc.  To  locate  the  centers  of 
the  six  small  circles  within  the  circle  HPT  R,  draw  a  circle 
with  a  radius  of  -1-^  inch  and  having  the  center  in  I.  The  small 
circles  have  a  radius  of  ^  inch. 

Inking.  In  inking  this  plate,  the  outlines  of  the  squares  of 
the  various  figures  are  inked  only  in  Figs.  2  and  3.  In  Fig.  1  the 
only  lines  to  be  inked  are  those  shown  in  full  lines  in  PLAT K 
III.  First  ink  the  star  and  then  the  square  and  diamonds.  The 
cross  hatching  should  be  done  without  measuring  the  distance  be- 
tween the  lines  and  without  the  aid  of  any  cross  hatching  device 
as  this  is  an  exercise  for  practice.  The  lines  should  be  about  ^ 
inch  apart.  After  inking  erase  all  construction  lines. 

In  inking  Fig.  2  be  careful  not  to  run  over  lines.  Each 
line  should  coincide  with  the  pencil  line.  The  student  should 


251 


38  MECHANICAL  DRAWING. 

first  ink  the  horizontal  lines  L,  M  and  K  and  the  vertical  lines 
E,  F  and  G.  The  short  lines  should  have  the  same  width 
but  the  border  lines,  A  B,  B  C,  C  D  and  D  A  should  be  a 
little  heavier. 

Fig.  3  is  drawn  entirely  with  the  45-degreG  triangle.  In  ink- 
ing the  oblique  lines  make  P  I,  R  K,  T  M,  etc.,  a  light  distinct 
line.  The  alternate  lines  O  II,  Q  J,  S  L,  etc.,  should  be  some- 
what heavier.  All  of  the  lines  which  slope  in  the  opposite  direc- 
tion are  light.  After  inking  Fig.  3  all  horizontal  and  vertical 
lines  (except  the  border  lines)  should  be  erased.  The  border 
lines  should  be  slightly  heavier  than  the  light  oblique  lines. 

The  only  instrument  used  in  inking  Fig.  4  is  the  compasses. 
In  doing  this  exercise  adjust  the  legs  of  the  compasses  so  that  the 
pen  will  always  be  perpendicular  to  the  paper.  If  this  is  not 
done  both  nibs  will  not  touch  the  paper  and  the  line  will  be  ragged. 
In  inking  the  arcs,  see  that  the- pen  stops  exactly  at  the  diagonals. 
The  circle  passing  through  T  and  the  small  inner  circle  should  be 
dotted  as  shown  in  PLATE  ///.  After  inking  the  circles  and 
arcs  erase  the  construction  lines  that  are  without  the  outer  circles 
but  leave  in  pencil  the  diagonals  inside  the  circle. 

In  Fig.  5  draw  all  arcs  first  and  then  draw  the  straight  lines 
meeting  these  arcs.  It  is  much  easier  to  draw  straight  lines  meet- 
ing arcs,  or  tangent  to  them,  than  to  make  the  arcs  tangent  to 
straight  lines.  As  this  exercise  is  difficult,  and  in  all  mechanical 
and  machine  drawing  arcs  and  tangents  are  frequently  used  we 
advise  the  beginner  to  draw  this  exercise  several  times.  Leave 
all  construction  lines  in  pencil. 

Fig.  6,  like  Fig.  4,  is  an  exercise  with  compasses.  If  Fig.  6 
has  been  laid  out  accurately  in  pencil,  the  inked  arcs  will  be  tan- 
gent to  each  other  and  the  finished  exercise  will  have  a  good 
appearance.  If,  however,  the  distances  were  not  accurately 
measured  and  the  lines  carefully  drawn  the  inked  arcs  will  not  be 
tangent.  The  arcs  whose  centers  are  L,  F,  JVI  and  E  and  A,  B,  C 
and  D  should  be  heavier  than  the  rest.  The  small  circles  may  be 
drawn  with  the  bow  pen.  After  inking  the  arcs  all  construction 
lines  should  be  erased. 


252 


MECHANICAL  DRAWING. 

PART     II. 
QEOflETRICAL  DEFINITIONS. 

A  point  is  used  for  marking  position ;  it  lias  neither  length, 
breadth  nor  thickness. 

A  line  has  length  only ;  it  is  produced  by  the  motion  of  a 
point. 

A  straight  line  or  right  line  is  one  that  has  the  same  direction 
throughout.  It  is  the  shortest  distance  between  any  two  of  its 
points. 

A  curved  line  is  one  that  is  constantly  changing  in  direction. 
It  is  sometimes  called  a  curve. 

A  broken  line  is  one  made  up  of  several  straight  lines. 

Parallel  lines  are  equally  distant  from  each  other  at  all 
points. 

A  horizontal  line  is  one  having  the  direction  of  a  line  drawn 
upon  the  surface  of  water  that  is  at  rest.  It  is  a  line  parallel  to 
the  horizon. 

A  vertical  line  is  one  that  lies  in  the  direction  of  a  thread 
suspended  from  its  upper  end  and  having  a  weight  at  the  lower 
end.  It  is  a  line  that  is  perpendicular  to  a  horizontal  plane. 

Lines  are  perpendicular  to  each  other,  if  when  they  cross, 
the  four  angles  formed  are  equal.  If  they  meet  and  form  two 
equal  angles  they  are  perpendicular. 

An  oblique  line  is  one  that  is  neither  vertical  nor  horizontal. 

In  Mechanical  Drawing,  lines  drawn  along  the  edge  of  the 
T  square,  when  the  head  of  the  T  square  is  resting  against  the 
left-hand  edge  of  the  board,  are  called  horizontal  lines.  Those 
drawn  at  right  angles  or  perpendicular  to  the  edge  of  the  T  square 
are  called  vertical. 

If  two  lines  cut  each  other,  they  are  called  intersecting1  line»y 
and  the  point  at  which  they  cross  is  called  the  point  of  intervention. 

Copyright,  J90t>,  by  Amtrjcan  School  of  Co 


855 


40  MECHANICAL  DRAWING. 

ANGLES. 

An  angle  is  formed  when  two  straight  lines  meet.  An  angle 
is  often  defined  as  being  the  difference  in  direction  of  two  straight 
lines.  The  lines  are  called  the  sides  and  the  point  of  meeting  is 
called  the  vertex.  The  size  of  an  angle  depends  upon  the  amount 
of  divergence  of  the  sides  and  is  independent  of  the  length  of 
these  lines. 


BIGHT   ANGLE.  ACUTE  ANGLE.  OBTUSE  ANGLE. 

If  one  straight  line  meet  another  and  the  angles  thus  formed 
are  equal  they  are  right  angles.  When  two  lines  are  perpendic- 
ular to  each  other  the  angles  formed  are  right  angles. 

An  acute  angle  is  less  than  a  right  angle. 

An  obtuse  angle  is  greater  than  a  right  angle. 

SURFACES. 

A  surface  is  produced  by  the  motion  of  a  line;  it  has  two 
dimensions,  —  length  and  breadth. 

A  plane  figure  is  a  plane  bounded  on  all  sides  by  lines;  the 
space  included  within  these  lines  (if  they  are  straight  lines)  is 
called  &  polygon  or  a  rectilinear  figure. 

TRIANGLES. 

A  triangle  is  a  figure  enclosed  by  three  straight  lines.  It  is 
a  polygon  of  three  sides.  The  bounding  lines  are  the  sides,  and 
the  points  of  intersection  of  the  sides  are  the  vertices.  The  angles 
of  a  triangle  are  the  angles  formed  by  the  sides. 

A  right-angled  triangle,  often  called  a  right  triangle,  is  one 
that  has  a  right  angle. 

An  acute-angled  triangle  is  one  that  has  all  of  its  angles  acute, 

An  obtuse-angled  triangle  is  one  that  has  an  obtuse  angle. 

In  an  equilateral  triangle  all  of  the  sides  are  equal. 


350 


MECHANICAL  DRAWING. 


•11 


If  all  of  the  angles  of  a  triangle  are  equal,  the  figure  is  called 
an  equiangular  triangle. 

A  triangle  is  called  scalene,  when  no  two  of  its  sides  are 
equal. 

In  an  isosceles  triangle  two  of  the  sides  are  equal. 


BIGHT  ANGLED  TRIANGLE.      ACUTE  ANGLED  TRIANGLE. 


OBTUSE  ANGLED  TRIANGLE. 


The  base  of  a  triangle  is  the  lowest  side  ;  however,  any  side 
may  be  taken  as  the  base.  In  an  isosceles  triangle  the  side  which 
is  not  one  of  the  equal  sides  is  usually  considered  the  base. 

The  altitude  of  a  triangle  is  the  perpendicular  drawn  from 
the  vertex  to  the  base. 


EQUILATERAL  TRIANGLE.  ISOSCELES   TRIANGLE.  SCALENE  TRIANGLE. 

QUADRILATERALS. 

A  quadrilateral  is  a  plane  figure  bounded  by  four  straight 
lines. 

The  diagonal  of  a  quadrilateral  is  a  straight  line  joining  two 
opposite  vertices. 


QUADRILATERAL. 


PARALLELOGRAM. 


A  trapezium  is  a  quadrilateral,  no  two  of  whose  sides  are 
parallel. 

A  trapezoid  is  a  quadrilateral  having  two  sides  parallel 


257 


42 


MECHANICAL  DRAWING. 


The  bases  of  a  trapezoid  are  its  parallel  sides.  The  altitude 
is  the  perpendicular  distance  between  the  bases. 

A  parallelogram  is  a  quadrilateral  whose  opposite  sides  are 
parallel. 

The  altitude  of  a  parallelogram  is  the  perpendicular  distance 
between  the  bases  which  are  the  parallel  sides. 

There  are  four  kinds  of  parallelograms: 


RECTANGLE. 


A  rectangle  is  a  parallelogram,  all  of  whose  angles  are  right 
angles.  The  opposite  sides  are  equal. 

A  square  is  a  rectangle,  all  of  whose  sides  are  equal. 

A  rhombus  is  a  parallelogram  which  has  four  equal  sides; 
but  the  angles  are  not  right  angles. 

A  rhomboid  is  a  parallelogram  whose  adjacent  sides  are 
anequal ;  the  angles  are  not  right  angles. 

POLYGONS. 

A  polygon  is  a  plane  figure  bounded  by  straight  lines. 
The  boundary  lines  are  called  the  sides  and  the  sum  of  the 
sides  is  called  the  perimeter. 

Polygons  are  classified  according  to  the  number  of  sides. 

A  triangle  is  a  polygon  of  three  sides. 

A  quadrilateral  is  a  polygon  of  four  sides. 

A  pentagon  is  a  polygon  of  five  sides. 

A  hexagon  is  a  polygon  of  six  sides. 

A  heptagon  is  a  polygon  of  seven  sides. 

An  octagon  is  a  polygon  of  eight  sides. 

A  decagon  is  a  polygon  of  ten  sides. 

A  dodecagon  is  a  polygon  of  twelve  sides. 
An  equilateral  polygon  is  one  all  of  whose  sides  are  equal. 
An  equiangular  polygon  is  one  all  of  whose  angles  are  equal. 
A  regular  polygon  is  one  all  of  whose  angles  are  equal  and  all 
sf  whose  sides  are  equal. 


258 


MECHANICAL  DRAWING. 


43 


CIRCLES. 

A  circle  is  a  plane  figure  bounded  by  a  curved  line,  every  point 
&f  which  is  equally  distant  from  a  point  within  called  the  center. 

The  curve  which  bounds  the  circle  is  called  the  circumference 
Any  portion  of  the  circumference  is  called  an  arc. 

The  diameter  of  a  circle  is  a  straight  line  drawn  through  the 
center  and  terminating  in  the  circumference.  A  radius  is  a 
straight  line  joining  the  center  with  the  circumference.  It  has  a 
length  equal  to  one  half  the  diameter.  All  radii  (plural  of 
radius)  are  equal  and  all  diameters  are  equal  since  a  diameter 
equals  two  radii. 


An  arc  equal  to  one-half  the  circumference  is  called  a  semi- 
circumference^  and  an  arc  equal  to  one-quarter  of  the  circumfer- 
ence is  called  a  quadrant.  A  quadrant  may  mean  the  sector,  arc 
or  angle. 

A  chord  is  a  straight  line  joining  the  extremities  of  an  arc. 
It  is  a  line  drawn  across  a  circle  that  does  not  pass  through  the 
center. 

A  secanjt  is  a  straight  line  which  intersects  the  circumference 
in  two  points. 


CIRCLE. 


A  tangent  is  a  straight  line  which  touches  the  circumference 
at  only  one  point.  It  does  not  intersect  the  circumference.  The 
point  at  which  the  tangent  touches  the  circumference  is  called  the 
point  of  tangency  or  point  of  contact. 


359 


MECHANICAL  DRAWING. 


A  sector  of  a  circle  is  the  portion  or  area  included  between 
an  arc  and  two  radii  drawn  to  the  extremities  of  the  arc. 

A  segment  of  a  circle  is  the  area  included  between  an  arc 
and  its  chord. 

Circles  are  tangent  when  the  circumferences  touch  at  only 
one  point  and  are  concentric  when  they  have  the  same  center. 


CONCENTRIC   CIRCLES. 


INSCRIBED  POLYGON 


An  inscribed  angle  is  an  angle  whose  .vertex  lies  in  the  cir- 
cumference and  whose  sides  are  chords.  It  is  measured  by  one- 
half  the  intercepted  arc. 

A  central  angle  is  an  angle  whose  vertex  is  at  the  center  of 
the  circle  and  whose  sides  are  radii. 


An  inscribed  polygon  is  one  whose  vertices  lie  in  the  circum- 
ference and  whose  sides  are  chords. 


MEASUREflENT  OF  ANGLES. 

To  measure  an  angle  describe  an  arc  "with  the  center  at  the 
vertex  of  the  angle  and  having  any  convenient  radius.  The  por- 
tion of  the  arc  included  between  the  sides  of  the  angle  is  the 
measure  of  the  angle.  If  the  arc  has  a  constant  radius  the  greater 
the  divergence  of  the  sides,  the  longer  will  be  the  arc.  If  there 
are  several  arcs  drawn  with  the  same  center,  the  intercepted  arcs 
will  have  different  lengths  bub  they  will  all  be  the  same  fraction 
of  the  entire  circumference. 

In  order  that  the  size  of  an  angle  or  arc  may  be  stated  with- 


MECHANICAL  DRAWING. 


45 


7J 


out  saying  that  it  is  a  certain  fraction  of  a  circumference,  the  cir- 
cumference is  divided  into  360 
equal  parts  called  degrees.  Thus 
we  can  say  that  an  angle  contains 
45  degrees,  which  means  that  it  is 
jfg5o"  —  |  of  a  circumference.  In 
order  to  obtain  accurate  measure- 
ments each  degree  is  divided  into 
60  equal  parts  called  minutes  and 
each  minute  is  divided  into  60  equal 
parts  called  seconds.  Angles  and 
arcs  are  usually  measured  by  means  of  an  instrument  called 
protractor  which  has  already  been  explained. 


/eo 


SOLIDS. 

A  polyedron  is  a  solid  bounded  by  planes.  The  bounding 
planes  are  called  the  faces  and  their  intersections  edges.  The 
intersections  of  the  edges  are  called  vertices. 

A  polygon  having  four  faces  is  called  a  tetraedron ;  one  having 
six  faces  a  hexaedron ;  of  eight  faces  an  octaedron ;  of  twelve 
faces  a  dodecaedron,  etc. 


RIGHT  PRISM. 


TRUNCATED    PRISM. 


A  prism  is  a  polyedron,  of  which  two  opposite  faces,  called 
bases,  are  equal  and  parallel ;  the  other  faces,  called  lateral  faces 
are  parallelograms. 

The  area  of  the  lateral  faces  is  called  the  lateral  area. 

The  altitude  of  a  prism  is  the  perpendicular  distance  between 
the  bases. 

Prisms  are  triangular,  quadrangular,  etc.,  according  to  the 
shape  of  the  base. 

A  right  prism  js  one  whose  lateral  edges  are  perpendicular 
to  the  bases. 


261 


46 


MECHANICAL  DRAWING. 


A  regular  prism  is  a  right  prism  having  regular  polygons  for 


A  parallelepiped  is  a  prism  whose  bases  are  parallelograms. 
If  the  edges  are  all  perpendicular  to  the  bases  it  is  called  a  right 
parallelepiped. 

A  rectangular  parallelepiped  is  a  right  parallelepiped  whose 
bases  are  rectangles  ;  all  the  f  ices  are  rectangles. 


PAKALLELOP1PKD. 


OCTAEDRON. 


A  cube  is  a  rectangular  parallelepiped  all  of  whose  faces  are 
squares. 

A.  truncated  prism  is  the  portion  of  a  prism  included  between 
the  base  and  a  plane  not  parallel  to  the  base. 

PYRAMIDS. 

A  pyramid  is  a  polyedron  one  face  of  which  is  a  polygon 
(called  the  base)  and  the  other  faces  are  triangles  having  a  com- 
mon vertex. 


REGULAR  PYRAMID. 


FRUSTUM  OF  PYRAMID. 


The  vertices  of  the  triangles  form  the  vertex  of  the  pyramid. 

The  altitude  of  the  pyramid  is  the  perpendicular  distance 
from  the  vertex  to  the  base. 

A  pyramid  is  called  triangular,  quadrangular,  etc.,  accord- 
ing to  the  shape  of  .the  base. 

A.  regular  pyramid  is  one  whose  base  is  a  regular  polygon 


MECHANICAL  DRAWING. 


17 


and  whose  vertex  lies  in  the  perpendicular  erected  at  the  center 
of  the  base. 

A  truncated  pyramid  is  the  portion  of  a  pyramid  included 
between  the  base  and  a  plane  not  parallel  to  the  base. 

A  frustum  of  a  pyramid  is  the  solid  included  between  the 
base  and  a  plane  parallel  to  the  base. 

The  altitude  of  a  frustum  of  a  pyramid  is  the  perpendicular 
distance  between  the  bases. 

CYLINDERS. 

A  cylindrical  surface  is  a  curved  surface  generated  by  the 
motion  of  a  straight  line  which  touches  a  curve  and  continues 
parallel  to  itself. 

A  cylinder  is  a  solid  bounded  by  a  cylindrical  surface  and 
two  parallel  planes  intersecting  this  surface. 

The  parallel  faces  are  called  bases. 


CYLINDER. 


RIGHT   CYLINDER 


-UIHEl)  CYLINDER. 


The  altitude  of  a  cylinder  is  the  perpendicular  distance 
between  the  bases. 

A  circular  cylinder  is  a  cylinder  whose  base  is  a  circle. 

A  right  cylinder  or  a  cylinder  of  revolution  is  a  cylinder  gen- 
erated by  the  revolution  of  a  rectangle  about  one  side  as  an  axis. 

A  prism  whose  base  is  a  regular  polygon  may  be  inscribed  in 
or  circumscribed  about  a  circular  cylinder. 

The  cylindrical  area  is  call  the  lateral  area.  The  total  area 
is  the  area  of  the  bases  added  to  the  lateral  area. 

CONES. 

A  conical  surface  is,  a  curved  surface  generated  by  the 
motion  of  a  straight  line,  one  point  of  which  is  fixed  and  the  end 
On  ends  of  which  move  in  a  curve. 


48 


MECHANICAL  DRAWING. 


A  cone  is  a  solid  bounded  by  a  conical  surface  and  a  plane 
which  cuts  the  conical  surface. 

The  plane  is  called  the  base  and   the   curved  surface   the 
lateral  area. 

.    The  vertex  is  the  fixed  point. 

The  altitude  of  a  cone  is  the  perpendicular  distance  from  the 
vertex ^to  the  base.. 

An  element  of  a  cone  is  a  straight  line  from  the  vertex  to  the 
perimeter  of  the  base. 

A  circular  cone  is  a  cone  whose  base  is  a  circle. 


RIGHT  CIRCULAR   CONE. 


FRUSTUM  OF  CONE. 


A  right  circular  cone  or  cone  of  revolution  is  a  cone  whose 
axis  is  perpendicular  to  the  base.  It  may  be  generated  by  the 
revolution  of  a  right  triangle  about  otoe  of  the  perpendicular  sides 


as  an  axis. 


A  frustum  of  a  cone  is  the  solid  included  between  the  base 
and  a  plane  parallel  to  the  base. 


TANGENT  PLANE. 


Tho  altitude  of  a  frustum  of  a  cone    is   the  perpendicular 
distance  between  the  bases. 

SPHERES. 

A  sphere  is  a  solid  bounded  by  a  curved  surface,  every  point 

of  which  is  equally  distant  from  a  point  within  called  the  center. 

The  radius  of  a  sphere  is  a  straight  line  drawn  from  the 


264 


MECHANICAL  DRAWING. 


center   to  the  surface.     The  diameter  is  a  straight   line  drawn 
through  the  center  and  having  its  extremities  in  the  surface. 

A  sphere  may  be  generated  by  the  revolution  of  a  semi-circle 
about  its  diameter  as  an  axis. 

An  inscribed  polyedron  is  a  polyedron  whose  vertices  lie  in 
the  surface  of  the  sphere. 

An  circumscribed  polyedron  is  a  polyedron  whose  faces  are 
tangent  to  a  sphere. 

•  A  great  circle  is  the  intersection  of  the  spherical  surface  ana 
a  plane  passing  through  thj  center  of  a  sphere. 

A  small  circle  is  the  intersection  of  the  spherical  surface  and 
a  plane  which  does  not  pass  through  the  center. 

A  sphere  is  tangent  to  a  plane  when  the  plane  touches  the 
surface  in  only  one  point.  A  plane  perpendicular  to  the  extremity 
of  a  radius  is  tangent  to  the  sphere. 

CONIC  SECTIONS. 

If  a  plane  intersects  a  cone  the  geometrical  figures  thus 
formed  are  called  conic  sections.  A  plane  perpendicular  to  the 
base  and  passing  through  the  vertex  of  a  right  circular  cone  forms 
an  isosceles  triangle.  If  the  plane  is  parallel  to  the  base  the 
intersection  of  the  plane  and  conical  surface  will  be  the  circum- 
ference of  a  circle. 


Fig.  1.  Fig.  2.  Fig.  3.  Fig.  4. 

Ellipse.  The  ellipse  is  a  curve  formed  by  the  intersection  of 
a  plane  and  a  cone,  the  plane  being  oblique  to  the  axis  but  not 
cutting  the  base.  If  a  plane  is  passed  through  a  cone  as  shown 
in  Fig.  1  or  through  a  cylinder  as  shown  in  Fig  2,  the  curve  of 
intersection  will  be  an  ellipse.  An  ellipse  may  be  denned  as 
being  a  curve  generated  by  a  point  moving  in  a  plane,  the  sum  oj 
the  distances  of  the  point  to  two  fixed  points  being  always  constant. 

The  two  fixed  points  are  called   the  foci  and   lie   on   the 


265 


50 


MECHANICAL  DRAWING. 


longest  line  that  can  be  drawn  in  the  ellipse.  One  of  these  points 
is  called  a  focus. 

The  longest  line  that  can  be  drawn  in  an  ellipse  is  called  the 
major  axis  and  the  shortest  line,  passing  through  the  center,  is 
called  the  minor  axis.  The  minor  axis  is  perpendicular  to  the 
middle  point  of  the  major  axis  and  the  point  of  intersection  is 
called  the  center 

An  ellipse  may  be  constructed  if  the  major  and  minor  axes 
are  given  or  if  the' foci  and  one  axis  are  known. 


PAKABOLA. 


Parabola.  The  parabola  is  a  curve  formed  by  the  inter- 
section of  a  cona  and  a  plane  parallel  to'  an  element  as  shown  in 
Fig.  3.  The  curve  is  not  a  closed  curve.  The  branches  approach 
parallelism. 

A  parabola  may  be  defined  as  l>eing  a  curve  every  point  of 
which  is  equally  distant  from  a  line 
and  a  point. 

The  point  is  called  the  focus  and 
the  given  line  the  directrix.  The 
line  perpendicular  to  the  directrix 
and  passing  through  the  focus  is 
the  axis.  The  intersection  of  the 
axis  and  the  curve  is  the  vertex. 

Hyperbola.  This  curve  is  formed 

by  the  intersection  of  a  plane  and  a  cone,  the  plane  being  parallel 
to  the  axis  of  the  cone  as  shown  in  Fig.  4.  Like  the  parabola, 
the  curve  is  not  a  closed  curve ;  the  branches  constantly  diverge. 
An  hyperbola  is  defined  as  being  a  plane  curve  such  that  the 
difference  of  the  distances  from  any  point  in  the  curve  to  two  fixed 
points  is  etjual  to  a  given  distance. 


HYPERBOLA. 


266 


MECHANICAL  DRAWING. 


51 


The  two  fixed  points  are  the  foci  and  the  line  passing  through 
them  is  the  transverse  axis. 

Rectangular  Hyperbola.  The  form  of  hyperbola  most  used 
in  Mechanical  Engineering  is  called  the  rectangular  hyperbola 
because  it  is  drawn  with  reference  to  rectangular  co-ordinates. 
This  curve  is  constructed  as  follows  :  In  Fig.  5,  O  X  and  O  Y  are 
the  two  co-ordinates  drawn  at  right  angles  to  each  other.  These 
lines  are  also  called  axes  or 
asymptotes.  Assume  A  to 
be  a  known  point  on  the 
curve.  In  drawing  this  curve  M 
for  the  theoretical  indicator 
card,  this  point  A  is  the  point 
of  cut-off. 

Draw  A  C  parallel  to 
O  X  and  A  D  perpendicular 
to  O  X.  Now  mark  off  any 
convenient  points  on  A  C  such  as  E,  F,  G,  and  H  ;  and  through 
these  points  draw  EE',  FF',  GG',  and  HH'  perpendicular  to  O  X. 
Connect  E,  F,  G,  H  anoLC  with  O.  Through  the  points  of  inter- 
section of  the  oblique  lines  and  the  vertical  line  A  D  draw  the 
horizontal  lines  LL',  MM',  NN',  PP'  and  QQ'.  The  first  point  on 
the  curve  is  the  assumed  point  A,  the  second  point  is  R,  the 
intersection  of  LL'  and  EE'.  The  third  is  the  intersection  S 
of  MM'  and  FF';  the  fourth  is  the  intersection  T  of  NN'  and 
GG'.  The  other  points  are  found  in  the  same  way. 

In  this  curve  the  products  of  the  co-ordinates  of  all  points  are 
equal.  Thus  LR  X  RE'  =  MS  X  SF'=  NT  X  TG'. 

ODONTOIDAL  CURVES. 

The  outlines  of  the  teeth  of  gears  must  be  drawn  accurately 
because  the  smoothness  of  running  depends  upon  the  shape  of  the 
teeth.  The  two  classes  of  curves  generally  employed  in  drawing 
gear  teeth  are  the  cycloidal  and  involute. 

Cycloid.  The  cycloid  is  a  curve  generated  by  a  point  on  the 
circumference  of  a  circle  which  rolls  on  a  straight  line  tangent  to 
the  circle. 

The  rolling  circle  is  called  the  describing  or  generating  circle 


52 


MECHANICAL  DRAWING. 


and  the  point,  the  describing  or  generating  point.     The    tangent 
along  which  the  circle  rolls  is  called  the  director. 

In  order  that  the  curve  may  be  a  true  cycloid  the  circle  must 
roll  without  any  slipping. 


7XNGENT     OR    D/RECTOR 


Epicycloid.  If  the  generating  circle  rolls  upon  the  outride 
of  an  arc  or  circle,  called  the  director  circle,  the  curve  thus  gener- 
ated is  called  an  epicycloid.  The  method  of  drawing  this  curve 
is  the  same  as  that  for  the  cycloid. 

Hypocycloid.  In  case  the  generating  circle  rolls  upon  the 
inside  of  an  arc  or  circle,  the  curve  thus  generated  is  called  the 
hypocycloid.  The  circle  upon  which  the  generating  circle  rolls  is 


called  the  director  circle.  If  the  generating  circle  has  a  diameter 
equal  to  the  radius  of  the  director  circle  the  hypocycloid  becomes 
a  straight  line. 

Involute.  If  a  thread  or  fine  wire  is  wound  around  a 
cylinder  or  circle  and  then  unwound,  the  end  will  describe  a 
curve  called  an  involute.  The  involute  may  be  defined  as  being 
a  curve  generated  by  a  point  in  a  tangent  rolling  on  a  circle  known 
as  the  base  circle. 

The  construction  of  the  ellipse,  parabola,  hyperbola  and 
odontoidal  curves  will  be  taken  up  in  detail  with  the  plates. 


MECHANICAL  DRAWING.  53 

PLATE  IV. 

Pencilling.  The  horizontal  and  vertical  center  lines  and  the 
border  lines  for  PLATE  IV  should  te  laid  out  in  the  same 
manner  as  were  those  for  PLATE  I.  There  are  to  be  six  figures 
on  this  plate  and  to  facilitate  the  laying  out  of  the  work,  the  fol- 
lowing lines  should  be  drawn :  measure  off  2^  inches  on  botli  sides 
of  the  vertical  center  line  and  through  these  points  draw  vertical 
lines  as  shown  in  dot  and  dash  lines  on  PLATE  IV.  In  these 
six  spaces  the  six  figures  are  to  be  drawn,  the  student  placing 
them  in  the  centers  of  the  spaces  so  that  they  will  present  a  good 
appearance.  In  locating  the  figures,  they  should  be  placed  a  little 
above  the  center  so  that  there  will  be  sufficient  space  below  to 
number  the  problem. 

The  figures  of  the  problems  should  first  be  drawn  lightly  in 
pencil  and  after  the  entire  plate  is  completed  the  lines  should  be 
inked.  In  pencilling,  all  intersections  must  be  formed  with  great 
care  as  the  accuracy  of  the  results  depends  upon  the  pencilling. 
Keep  the  pencil  points  in  good  order  at  all  times  and  draw  lines 
exactly  through  intersections. 

GEOMETRICAL  PROBLEMS. 

The  following  problems  are  of  great  importance  to  the 
mechanical  draughtsman.  The  student  should  solve  them  with 
care ;  he  should  not  do  them  blindly,  but  should  understand  them 
so  that  he  can  apply  the  principles  in  later  work. 

PROBLEM  I.     To  Bisect  a  Given  Straight  Line. 

Draw  the  horizontal  straight  line  A  C  about  3  inches  long. 
With  the  extremity  A  as  a  center  and  any  convenient  radius 
(about  2  inches)  describe  arcs  above  and  below  the  line  A  C. 
With  the  other  extremity  C  as  a  center  and  with  the  same  radius 
draw  short  arcs  above  and  below  A  C  intersecting  the  first  arcs  at 
D  and  E.  The  radius  of  these  arcs  must  be  greater  than  one-half 
the  length  of  the  line  in  order  that  they  may  intersect.  Now 
draw  the  straight  line  D  E  passing  through  the  intersections  D 
and  E.  This  line  cuts  the  line  A  C  at  F  which  is  the  middle 

point. 

AF  =  FC 


54  MECHANICAL  DRAWING. 

Proof.  Since  the  points  D  and  E  are.  equally  distant  from 
A  and  C  a  straight  line  drawn  through  them  is  perpendicular  to 
A  C  at  its  middle  point  F. 

PROBLEM  2.  To  Construct  an  Angle  Equal  to  a  Given 
Angle. 

Draw  the  line  O  C  about  2  inches  long  and  the  line  O  A  of 
about  the  same  length.  The  angle  formed  by  these  lines  may  be 
any  convenient  size  (about  45  degrees  is  suitable).  This  angle 
A  O  C  is  the  given  angle. 

Now  draw  F  G  a  horizontal  line  about  2^  inches  long  and  let 
F  the  left-hand  extremity  be  the  vertex  of  the  angle  to  be 
constructed. 

With  O  as  a  center  and  any  convenient  radius  (about  11 
inches)  describe  the  arc  L  M  cutting  both  O  A  and  OC.  With 
F  as  a  center  and  the  same  radius  draw  the  indefinite  arc  O  Q. 
Now  set  the  compass  so  that  the  distance  between  the  pencil  and 
the  needle  point  is  equal  to  the  chord  L  M.  With  Q  as  a  center 
and  a  radius  equal  to  L  M  draw  an  arc  cutting  the  arc  O  Q  at  P. 
Through  F  and  P  draw  the  straight  line  F  E.  The  angle  E  F  G 
is  the  required  angle  since  it  is  equal  to  A  O  C. 

Proof.  Since  the  chords  of  the  arcs  L  M  and  P  Q  are  equal 
the  arcs  are  equal.  The  angles  are  equal  because  with  equal 
radii  equal  arcs  are  intercepted  by  equal  angles. 

PROBLEM  3.  To  Draw  Through  a  Given  Point  a  Line 
Parallel  to  a  Given  Line. 

First  Method.  Draw  the  horizontal  straight  line  A  C  about 
3J  inches  long  and  assume  the  point  P  about  1J  inches  above 
A  C.  Through  the  point  P  draw  an  oblique  line  F  E  forming 
any  convenient  angle  with  A  C.  (Make  the  angle  about  60 
degrees).  Now  construct  an  angle  equal  to  P  F  C  having  the 
vertex  at  P  and  one  side  the  line  E  P.  (See  problem  2). 
This  may  be  done  as  follows :  With  F  as  a  center  and  any  con- 
venient radius,  describe  the  arc  L  M.  With  the  same  radius 
draw  the  indefinite  arc  N  O  using  P  as  the  center.  With  N  as  a 
center  and  a  radius  equal  to  the  chord  L  M,  draw  an  arc  cutting 
the  arc  N  O  at  O.  Through  the  points  P  and  O  draw  a  straight 
line  which  will  be  parallel  to  A  C. 


r" 

N 


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U 


MECHANICAL  DRAWING.  55 

Proof.  If  two  straight  lines  are  cut  by  a  third  making  the 
corresponding  angles  equal,  the  lines  are  parallel. 

PROBLEM  4.  To  Draw  Through  a  Given  Point  a  Line 
Parallel  to  a  Given  Line. 

Second  Method.  Draw  the  straight  line  A  C  about  3*  inches 
long  and  assume  the  point  P  about  1  £  inches  above  A  C.  With 
?  as  a  center  and  any  convenient  radius  (about  2£  inches)  draw 
the  indefinite  arc  E  D  cutting  the  line  A  C.  Now  with  the  same 
radius  and  with  D  as  a  center,  draw  an  arc  P  Q.  Set  the  com- 
pass so  that  the  distance  between  the  needle  point  and  the  pencil 
is  equal  to  the  chord  P  Q.  With  D  as  a  center  and  a  radius 
equal  to  P  Q,'  describe  an  arc  cutting  the  arc  E  D  at  H.  A  line 
drawn  through  P  and  H  will  be  parallel  to  A  C. 

Proof.  Draw  the  line  Q  H.  Since  the  arcs  P  Q  and  H  D 
are  equal  and  have  the  same  radii,  the  angles  P  H  Q  and  H  Q  D 
are  equal.  Two  lines  are  parallel  if  the  alternate  interior  angles 
are  equal. 

PROBLEM  5.  To  Draw  a  Perpendicular  to  a  Line  from 
a  Point  in  the  Line. 

First  Method.     When  the  point  is  near  the  middle  of  the  line. 

Draw  the  horizontal  line  A  C  about  3^  inches  long  and 
assume  the  point  P  near  the  middle  of  the  line.  With  P  as  a 
center  and  any  convenient  radius  (about  1 J  inches)  draw  two  arcs 
cutting  the  line  A  C  at  E  and  F.  Now  with  E  and  F  as  centers 
and  any  convenient  radius  (about  2J  inches)  describe  arcs  inter- 
secting at  O.  The  line  O  P  will  be"perpendicular  to  A  C  at  P. 

Proof.  The  points  P  and  O  are  equally  distant  from  E  and 
F.  Hence  a  line  drawn  through  them  is  perpendicular  to  the 
middle  point  of  E  F  which  is  P. 

PROBLEM  6.  To  Draw  a  Perpendicular  to  a  Line  from 
a  Point  in  the  Line. 

Second  Method.     When  the  point  is  near  the  end  of  the  line. 

Draw  the  line  A  C  about  3£  inches  long.  Assume  the  given 
point  P  to  be  about  |  inch  from  the  end  A.  With  any  point  D 
as  a  center  and  a  radius  equal  to  D  P,  describe  an  arc,  cutting  A  C 
at  E.  Through  E  and  D  draw  the  diameter  E  O.  A  line  from 
O  to  P  is  perpendicular  to  A  C  at  P. 


273 


50  MECHANICAL  DRAWING. 

Proof.  The  angle  O  P  E  is  inscribed  in  a  semi-circle ;  hence 
it  is  a  right  angle,  and  the  sides  O  P  and  P  E  are  perpendicular 
to  each  other. 

After  completing  these  figures  draw  pencil  lines  for  the 
lettering.  The  words  « PL  A  TE  TV"  and  the  date  and  name 
should  be  placed  in  the  border,  as  in  preceding  plates.  To 
letter  the  words  u  Problem  1,"  "  Problem  2,"  etc,,  draw  horizontal 
lines  |  inch  above  the  horizontal  center  line  and  the  lower  border 
line.  Draw  another  line  ^  inch  above,  to  limit  the  height  of  the 
P,  b  and  I.  Draw  a  third  line  |-  inch  above  the  lower  line  as  a 
guide  line  for  the  tops  of  the  small  letters. 

Inking.  In  inking  PLATE  JVthe  figures  should  be  inked 
first.  The  line  A  C  of  Problem  1  should  be  a  full  line  as  it  is 
the  given  line  ;  the  arcs  and  line  I)  E,  being  construction  lines 
should  be  dotted.  In  Problem  2,  the  sides  of  the  angles  should 
be  full  lines.  Make  the  chord  L  M  and  the  arcs  dotted,  since 
as  before,  they  are  construction  lines. 

In  Problem  3,  the  line  A  C  is  the  given  line  and  P  O  is  the 
line  drawn  parallel  to  it.  As  E  F  and  the  arcs  do  not  form  a  part 
of  the  problem  but  are  merely  construction  lines,  drawn  as  an  aid 
in  locating  P  O,  they  should  be  dotted.  In  Problems  4,  5  and  6, 
the  assumed  lines  and  those  found  by  means  of  the  construction 
lines  should  be  full  lines.  The  arcs  and  construction  lines  should 
^  dotted.  In  Problem  6,  the  entire  circumference  need  not  be 
Inked,  only  that  part  is  necessary  that  is  used  in  the  problem. 
The  inked  arc  should  however  be  of  sufficient  length  to  pass 
through  the  points  O,  P  and  E. 

After  inking  the  figures,  the  border  lines  should  be  inked 
with  a  heavy  line  as  before.  Also,  the  words  "PLATE IV"  and 
the  date  and  the  student's  name.  Under  each  problem  the  words 
"Problem  1,"  "Problem  2,"  etc.,  should  be  inked;  lower  case  let- 
ters being  used  as  shown. 

PLATE  V. 

Pencilling.  In  laying  out  the  border  lines  and  centre  lines 
follow  the  directions  given  for  PLATE  IV.  The  dot  and 
dash  lines  should  be  drawn  in  the  same  manner  as  there  are  to  be 
six  problems  on  this  plate. 


274 


MECHANICAL  DRAWING.  57 

PROBLEM  7.  To  Draw  a  Perpendicular  to  a  Line  from  a 
Point  without  the  Line. 

Draw  the  horizontal  straight  line  A  C  about  3|  inches  long. 
Assume  the  point  P  about  li  inches  above  the  line.  With  P  as 
a  center  and  any  convenient  radius  (about  2  inches)  describe  an 
arc  cutting  A  C  at  E  and  F.  The  radius  of  this  arc  must  always 
be  such  that  it  will  cut  A  C  in  two  points;  the  nearer  the  points 
E  and  F  are  to  A  and  C,  the  greater  will  be  the  accuracy  of  the 
work.  Now  with  E  and  F  as  centers  and  any  convenient  radius 
(about  2^  inches)  draw  the  arcs  intersecting  below  A  C  at  T.  A 
line  through  the  points  P  and  T  will  be  perpendicular  to  A  C. 

In  case  there  is  not  room  below  A  C  to  draw  the  arcs,  they 
may  be  drawn  intersecting  above  the  line  as  shown  at  N.  When- 
ever convenient,  draw  the  arcs  below  A  C  for  greater  accuracy. 

Proof.  Since  P  and  T  are  equally  distant  from  E  and  F, 
the  line  P  T  is  perpendicular  to  A  C. 

PROBLEM  8.     To  Bisect  a  Given  Angle. 

First  Method.     When  the  sides  intersect. 

Draw  the  lines  O  C  and  O  A  forming  any  angle  (from  45  to 
60  degrees).  These  lines  should  be  about  3  inches  long.  With 
O  as  a  center  and  any  convenient  radius  (about  2  inches)  draw 
an  arc  intersecting  the  sides  of  the  angle  at  E  and  F.  With  E 
and  F  as  centers  and  a  radius  of  1|  or  1|  inches,  describe  short 
arcs  intersecting  at  I.  A  line  O  D,  drawn  through  the  points  O 
and  I,  bisects  the  angle. 

In  solving  this  problem  the  arc  E  F  should  not  be  too  near 
the  vertex  if  accuracy  is  desired. 

Proof.  The  central  angles  A  O  D  and  DOC  are  equal 
because  the  arc  E  F  is  bisected  by  the  line  O  D.  The  point  I  is 
equally  distant  from  E  and  F. 

PROBLEM  9.     To  Bisect  a  Given  Angle, 

Second  Method.     When  the  lines  do  not  intersect. 

Draw  the  lines  A  C  and  E  F  about  4  inches  long  and  in 'the 
positions  as  shown  on  PLATE  V.  Draw  A'  C'  and  E'  F'  parallel 
to  A  C  and  E  F  and  at  such  equal  distances  from  them  that 
they  will  intersect  at  O.  Now  bisect  the  angle  C'  O  F'  by 


58  MECHANICAL  DRAWING. 

the  method  of  Problem  8.  Draw  the  arc  G  H  and  with  G  and  H 
as  centers  draw  the  arcs  intersecting  at  R.  The  line  O  R  bisects 
the  angle. 

Proof.  Since  A'  C'  is  parallel  to  A  C  and  E'  F'  parallel  to 
E  F,  the  angle  C'  O  F'  is  equal  to  the  angle  formed  by  the  lines 
A  C  and  E  F.  Hence  as  O  R  bisects  angle  C'  O  F'  it  also  bisects 
the  angle  formed  by  the  lines  A  C  and  E  F. 

PROBLEM  10.  To  Divide  a  Given  Line  into  any  Number 
of  Equal  Parts. 

Let  A  C,  about  3|^  inches  long,  be  the  given  line.  Let  us 
divide  it  into  7  equal  parts.  Draw  the  line  A  J  at  least  4  inches 
long,  forming  any  convenient  angle  with  A  C.  On  A  J  lay  off, 
by  means  of  the  dividers  or  scale,  points  D,  E,  F,  G,  etc.,  each  ^  inch 
apart.  If  dividers  are  used  the  spaces  need  not  be  exactly  ^ 
inch.  Draw  the  line  J  C  and  through  the  points  D,  E,  F,  G,  etc., 
draw  lines  parallel  to  J  C.  These  parallels  will  divide  the  line 
A  C  into  7  equal  parts. 

Proof.  If  a  series  of  parallel  lines,  cutting  two  straight 
lines,  intercept  equal  distances  on  one  of  these  lines,  they  also 
intercept  equal  distances  on  the  other. 

PROBLEM  11.  To  Construct  a  Triangle  having  given  the 
Three  Sides. 

Draw  the  three  sides  as  follows : 

A  C,  2|  inches  long. 
E  F,  llf  inches  long. 
M  N,  2^  inches  long. 

Draw  R  S  equal  in  length  to  A  C.  With  R  as  a  center  and 
a  radius  equal  to  E  F  describe  an  arc.  With  S  as  a  center  and 
a  radius  equal  to  M  N  draw  an  arc  cutting  the  arc  previously 
drawn,  at  T.  Connect  T  with  R  and  S  to  form  the  triangle. 

PROBLEM  12.  To  Construct  a  Triangle  having  given 
One  Side  and  the  Two  Adjacent  Angles. 

Draw  the  line  M  N  3J  inches  long  and  draw  two  angles 
A  O  D  and  E  F  G.  Make  the  angle  A  O  D  about  30  degrees  and 
E  F  G  about  60  degrees. 

Draw  R  S  equal  in  length  to  M  N  and  at  R  construct  an 


276 


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MECHANICAL  DRAWING. 


59 


angle  equal  to  A  O  D.  At  S  construct  an  angle  equal  to  E  F  G 
by  the  method  used  in  Problem  2.  PLATE  V  shows  the  neies- 
sary arcs.  Produce  the  sides  of  the  angles  thus  constructed 
they  meet  at  T.  The  triangle  R  TS  will  be  the  required 
triangle. 

After  drawing  these  six  figures  in  pencil,  draw  the  pencil 
lines  for  the  lettering.  The  lines  for  the  words  -PLATE  V," 
date  and  name,  should  be  pencilled  as  explained  on  page  <>0 
The  words  "Problem  7,"  "Problem  8,"  etc.,  are  lettered  as  for 
PLATE  IV. 

Inking.  In  inking  PLATE  V,  the  same  principles  should 
be  followed  as  stated  with  PLATE  IV.  The  student  should 
apply  these  principles  and  not  make  certain  lines  dotted  just 
because  they  are  shown  dotted  in  PLATE  V. 

After  inking  the  figures,  the  border  lines  should  be  inked 
and  the  lettering  inked  as  already  explained  in  connection  with 
previous  plates. 

PLATE    VI. 

Pencilling.  Lay  out  this  plate  in  the  same  manner  as  the 
two  preceding  plates. 

PROBLEM  13.  To  describe  an  Arc  or  Circumference 
through  Three  Given  Points  not  in  the  same  straight  line. 

Locate  the  three  points  A,  B  and  C.  Let  the  distance 
between  A  and  B  be  about  2  inches  and  the  distance  between  A 
and  C  be  about  2|  inches.  Connect  A  and  B  and  A  and  C. 
Erect  perpendiculars  to  the  middle  points  of  \  B  and  A  C.  This 
may  be  done  as  explained  with  Problem  1.  With  A  and  B  as 
centers  and  a  radius  of  about  1|  inches,  describe  the  arcs  inter- 
secting at  I  and  J.  With  A  and  C  as  centers  and  with  a  radius 
of  about  1|  inches  draw  the  arcs,  intersecting  at  E  and  F.  Now 
draw  light  pencil  lines  connecting  the  intersections  I  and  J  and 
E  and  F.  These  lines  will  intersect  at  O. 

With  O  as  a  center  and  a  radius  equal  to  the  distance  O  A, 
describe  the  circumference'passing  through  A,  B  and  C. 

Proof.  The  point  O  is  equally  distant  from  A,  B  and  C, 
since  it  lies  in  the  perpendiculars  to  the  middle  points  of  A  B  and 


279 


60  MECHANICAL  DRAWING. 


A  C.     Hence  the  circumference  will  pass  through  A,  B  and  C. 

PROBLEM  14.      To  inscribe  a  Circle  in  a  given  Triangle. 

Draw  the  triangle  L  M  N  of  any  convenient  size.  M  N  may 
be  made  3^  inches,  L  M,  2|  inches,  and  L  N,  3J  inches.  Bisect 
the  angles  M  L  N  and  L  M  N.  The  bisectors  M  I  and  L  J  may 
be  drawn  by  the  method  used  in  Problem  8.  Describe  the  arcs 
A  C  and  E  F,  having  centers  at  L  and  M  respectively.  The  arcs 
intersecting  at  I  and  J  are  drawn  as  already  explained.  The 
bisectors  of  the  angles  intersect  at  O,  which  is  the  center  of  the 
inscribed  circle.  The  radius  of  the  circle  is  equal  to  the  perpen- 
dicular distance  from  O  to  one  of  the  sides. 

Proof.  The  point  of  intersection  of  the  bisectors  of  the 
angles  of  a  triangle  is  equally  distant  from  the  sides. 

PROBLEM  15.  To  inscribe  a  Regular  Pentagon  in  a  given 
Circle. 

With  O  as  a  center  and  a  radius  of  about  1£  inches,  describe 
the  given  circle.  With  the  T  square  and  triangles  draw  the  cen- 
ter lines  A  C  and  E  F.  These  lines  should  be  perpendicular  to 
each  other  and  pass  through  O.  Bisect  one  of  the  radii,  such  as 
O  C,  and  with  this  point  H  as  a  center  and  a  radius  H  E,  describe 
the  arc  E  P.  This  arc  cuts  the  diameter  A  C  at  P.  With  E  as 
a  center  and  a  radius  E  P,  draw  arcs  cutting  the  circumference 
at  L  and  Q.  With  the  same  radius  and  a  center  at  L,  draw  the 
arc,  cutting  the  circumference  at  M.  To  find  the  point  N,  use 
either  M  or  Q  as  a  center  and  the  distance  E  P  as  a  radius. 

The  pentagon  is  completed  by  drawing  the  chords  E  L,  L  M, 
M  N,  N  Q  and  Q  E. 

PROBLEM  16.  To  inscribe  a  Regular  Hexagon  in  a  given 
Circle. 

With  O  as  a  center  and  a  radius  of  1|  inches  draw  the  given 
circle.  With  the  T  square  draw  the  diameter  A  D.  With  D  as 
a  center,  and  a  radius  equal  to  O  D,  describe  arcs  cutting  the 
circumference  at  C  and  E.  Now  with  C  and  E  as  centers  and 
the  same  radius,  draw  the  arcs,  cutting  the  circumference  at  B 
and  F.  Draw  the  hexagon  by  joining  the  points  thus  formed. 

To  inscribe  a  regular  hexagon  in  a  circle   mark  off  chords 

O  O 

equal  in  length  to  the  radius. 


r 


MECHANICAL  DRAWING.  01 

To  inscribe  an  equilateral  triangle  in  a  circle  the  same  method 
may  be  used.  The  triangle  is  formed  by  joining  the  opposite 
vertices  of  the  hexagon. 

Proof.  The  triangle  O  C  D  is  an  equilateral  triangle  by 
construction.  Then  the  angle  C  O  D  is  one-third  of  two  right 
angles  and  one-sixth  of  four  right  angles.  Hence  arc  C  D  is  one- 
sixth  of  the  circumference  and  the  chord  is  a  side  of  a  regular 
hexagon. 

PROBLEM  17.  To  draw  a  line  Tangent  to  a  Circle  at  a 
given  point  on  the  circumference. 

With  O  as  a  center  and  a  radius  of  about  1^  inches  draw 
the  given  circle.  Assume  some  point  P  on  the  circumference 
Join  the  point  P  with  the  center  O  and  through  P  draw  a  line 
F  P  perpendicular  to  P  O.  This  may  be  done  in  any  one  of  several 
methods.  Since  P  is  the  extremity  of  O  P  the  method  given  in 
Problem  6  of  PLATE  IV,  may  be  used. 

Produce  P  O  to  Q.  With  any  center  C,  and  a  radius  C  P 
draw  an  arc  or  circumference  passing  through  P.  Draw  E  F  a 
diameter  of  the  circle  whose  center  is  C  and  through  F  and  P 
draw  the  tangent. 

Proof.  A  line  perpendicular  to  a  radius  at  its  extremity  is 
tangent  to  the  circle. 

PROBLEM  18.  To  draw  a  line  Tangent  to  a  Circle  from  a 
point  outside  the  circle. 

With  O  as  a  center  and  a  radius  of  about  1  inch  draw  the 
given  circle.  Assume  P  some  point  outside  of  the  circle  about 
2J  inches  from  the  center  of  the  circle.  Draw  a  straight  line 
passing  through  P  and  O.  Bisect  P  O  and  with  the  middle 
point  F  as  a  center  describe  the  circle  passing  through  P  and  O. 
Draw  a  line  through  P  and  the  intersection  of  the  two  circum- 
ferences C.  The  line  P  C  is  tangent  to  the  given  circle.  Simi- 
larly P  E  is  tangent  to  the  circle. 

Proof.  The  angle  P  C  O  is  inscribed  in  a  semi-circle  and 
hence  is  a  right  angle.  Since  P  C  O  is  a  right  angle  P  C  is  per- 
pendicular to  C  O.  The  perpendicular  to  a  radius  at  its  extremity 
is  tangent  to  the  circumference. 

Inking.     In  inking  PLATE  VI  the  same  method  should  be 


62  MECHANICAL  DRAWING. 

followed  as  in  previous  plates.     The  name  and  address  should  be 
lettered  in  inclined  Gothic  capitals  as  before.  » 

PLATE   VII. 

Pencilling.  PLATE  VII should  be  laid  out  in  the  same 
manner  as  previous  plates.  Six  problems  on  the  ellipse,  spiral, 
parabola  and  hyperbola  are  to  be  constructed  in  the  six  spaces. 

PROBLEM  19.  To  draw  an  Ellipse  when  the  Axes  are 
given. 

Draw  the  lines  L  M  and  C  D  about  3^  and  2^  inches  long 
respectively.  Let  C  D  be  perpendicular  to  M  N  at  its  middle 
point  P.  Make  C  P  =  P  D.  These  two  lines  are  the  axes.  With 
C  as  a  center  and  a  radius  equal  to  one-half  the  major  axis  or 
equal  to  L  P,  draw  the  arc,  cutting  the  major  axis  at  E  and  F. 
These  two  points  are  the  foci.  Now  mark  off  any  convenient 
distances  on  P  M,  such  as  A,  B  and  G. 

With  E  as  a  center  and  a  radius  equal  to  L  A,  draw  arcs 
above  and  below  L  M.  With  F  as  a  center,  and  a  radius  equal 
to  A  M  describe  short  arcs  cutting  those  already  drawn  as  shown 
at  N.  With  E  as  a  center  and  a  radius  equal  to  L  B  draw  arcs 
above  and  below  L  M  as  before.  With  F  as  a  center  and  ?-  radius 
equal  to  B  M,  draw  arcs  intersecting  those  already  drawn  as  shown 
at  O.  The  point  P  and  others  are  found  by  repeating  the  process. 
The  student  is  advised  to  find  at  least  12  points  on  the  curve  — 
6  above  and  6  below  L  M.  These  12  points  with  L,  C,  M  and 
D  will  enable  the  student  to  draw  the  curve. 

After  locating  these  points,  a  free  hand  curve  passing  through 
them  should  be  sketched. 

PROBLEM  20.  To  draw  an  Ellipse  when  the  two  Axes  are 
given. 

Second  Method.  Draw  the  two  axes  A  B  and  P  Q  in  the 
same  manner  as  for  Problem  19.  With  O  as  a  center  and  a  radius 
equal  to  one-half  the  major  axis,  describe  the  circumference  A  C 
D  E  F  B.  Similarly  with  the  same  center  and  a  radius  equal  to 
one-half  the  minor  axis,  describe  a  circle.  Draw  any  radii  such 
as  O  C,  O  D,  O  E,  O  F,  etc.,  cutting  both  circumferences.  These 
radii  may  be  drawn  with  the  60  and  45  degree  triangles.  At  the 


L 


J 


MECHANICAL  DRAWING. 


c.:; 


pom*  of  mtersection  of  the  radii  with  the  Lug.  circle  C  D  E  and 
*»*"*•!  lines  and  from  the  intersection  of  the  radii  with 
circle  C',  D',  E',  and  F',  draw  lor^ntal  lines  intersect 


eS 


As  in  Problem  19,  a  free  hand  curve  should  be  sketched  pass- 
ing  through  these  points.     About  five   points  in  each  quadrant 

will  be  sufficient. 

PROBLEM  21.    To    draw   an    EI,ipse    by    means    of    . 
1  rammel. 

As  in  the  two  preceding  problems,  draw  the  major  and  minor 
axes,  U  V  and  X  Y.  Take  a  slip  of  paper  having  a  straight 
edge  and  mark  off  C  B  equal  to  one-half  the  major  axis,  and  D  B 
one-half  the  minor  axis.  Place  the  slip  of  paper  in  various 
positions  keeping  the  point  D  on  the  major  axis  and  the  point  C 
on  the  minor  axis.  If  this  is  done  the  point  B  will  mark  various 
points  on  the  curve.  Find  as  many  points  as  necessary  and  sketeh 
the  curve. 

PROBLEM  22.     To  draw  a  Spiral  of  one  turn  in  a  circle. 

Draw  a  circle  with  the  center  at  O  and  a  radius  of  1  '  inches. 
Mark  off  on  the  radius  O  A,  distances  of  one-eighth  inch.  As 
3  A  is  1J  inches  long  there  will  be  12  of  these  distances.  Draw 
circles  through  these  points.  Now  draw  radii  O  B,  O  C,  O  D 
etc.  each  30  degrees  apart  (use  the  30  degree  triangle).  This 
will  divide  the  circle  into  12  equal  parts.  The  curve  starts  at  the 
center  O.  The  next  point  is  the  intersection  of  the  line  O  B  and 
the  first  circle.  The  third  point  is  the  mtersection  of  O  C  and 
the  second  circle.  The  fourth  point  is  the  intersection  of  O  D 
and  the  third  circle.  Other  points  are  found  in  the  same  way. 
Sketch  in  pencil  the  curve  passing  through  these  points. 

PROBLEM  23.  To  draw  a  Parabola  when  the  Abscissa  and 
Ordinate  are  given. 

Draw  the  straight  line  A  B  about  three  inches  long.  This 
line  is  the  axis  or  as  it  is  sometimes  called  the  abscissa.  At  A 
and  B  draw  lines  perpendicular  to  A  B.  Also  with  the  T  square 
draw  E  C  and  F  D,  1|  inches  above  and  below  A  B.  Let  A  be 


287 


64  MECHANICAL  DRAWING. 

the  vertex  of  the  parabola.  Divide  A  E  into  any  number  of 
equal  parts  and  divide  E  C  into  the  same  number  of  equal  parts. 
Through  the  points  of  division,  R,  S,  T,  U  and  V,  draw  horizontal 
lines  and  connect  L,  M,  N,  O  and  P,  with  A.  The  intersections 
of  the  horizontal  lines  with  the  oblique  lines  are  points  on  the 
curve.  For  instance,  the  intersection  of  A  L  and  the  line  V  is 
one  point  and  the  intersection  of  A  M  and  the  line  U  is  another. 
The  lower  part  of  the  curve  A  D  is  drawn  in  the  same 
manner. 

PROBLEM  24.  To  draw  a  Hyperbola  when  the  abscissa 
E  X,  the  ordinate  A  E  and  the  diameter  X  Y  are  given. 

Draw  E  F  about  3  inches  long  and  mark  the  point  X,  1  inch 
from  E  and  the  point  Y,  1  inch  from  X  With  the  triangle  and 
T  square,  draw  the  rectangles  A  B  D  C  and  O  P  Q  R  such  that 
A  B  is  1  inch  in  length  and  A  C,  3  inches  in  length.  Divide 
A  E  into  any  number  of  equal  parts  and  A  B  into  the  same  num- 
ber of  equal  parts.  Draw  L  X,  M  X  and  N  X ;  also  connect  T, 
U  and  V  with  Y.  The  first  point  on  the  curve  is  the  intersection 
A ;  the  next  is  the  intersection  of  T  Y  and  L  X  ;  the  third  the 
intersection  of  U  Y  and  M  X.  The  remaining  points  are  found 
in  the  same  manner.  The  curve  X  C  and  the  right-hand  curve 
P  Y  Q  are  found  by  repeating  the  process. 

Inking.  In  inking  the  figures  on  this  plate,  use  the  French 
or  irregular  curve  and  make  full  lines  for  the  curves  and  their 
axes.  The  construction  lines  should  be  dotted.  Ink  in  all  the 
construction  lines  used  in  finding  one-half  of  a  curve,  and  in 
Problems  19,  20,  23  and  24  leave  all  construction  lines  in  pencil 
except  those  inked.  In  Problems  21  and  22  erase  all  construction 
lines  not  inked.  The  trammel  used  in  Problem  21  may  be  drawn 
in  the  position  as  shown,  or  it  may  be  drawn  outside  of  the  ellipse 
in  any  convenient  place. 

The  same  lettering  should  be  done  on  this  plate  as  on  previous 
plates. 

PLATE  VIII. 

Pencilling.  In  laying  out  Plate  VIII,  draw  the  border  lines 
and  horizontal  and  vertical  center  lines  as  in  previous  plates,  to 
divide  the  plate  into  four  spaces  for  the  four  problems. 


MECHANICAL  DRAWING. 



a  Cycloid  when  the  diameter 


zontal  lines.     Now  with  the  dividers  Lt'so'  thn 

between  the  points  is  equal  to  the  chord  of  the  arc  C  I)   nruk  off- 

'  p  on  the  line  A  B  --      " 


CO'      T  PerPe»dicu1^  to  the  center  line 

This  center  line  is  drawn  through  the  point  O'  with  the 
T  square  and  is  the  line  of  centers  of  the  generating  circle  as  it 
rolls  along  the  line  A  B.     Now  with  the  intersections  Q,  R   S 
I,  etc    of  these  verticals  with  the  center  line  as  centers  describe 
arcs  of  circles  as  shown.     The  points  on  the  curve  are  the  inter- 
sections of  these  arcs  and  the  horizontal  lines  drawn  through  the 
points  C,  D,  E,  F,  etc.     Thus  the  intersection  of  the  arc  whose 
•ter  ,s  Q  and  the  horizontal  line  through  C  is  a  point  I  on  the 
curve.     Similarly,  the  intersection  of  the  arc  whose  center  is  K 
and  the  horizontal  line  through  D  is  another  point  J  on  the  curve 
The  remaining  points,  as  well  as  those  on  the  right-hand  side   are 
found  in  the  same  manner.      To  obtain  great    accuracy  in  'this 
eurve,  the  circle  should  be  divided  into  a  large  number  of  equal 
parts,  because  the  greater  the  number  of  divisions  the  less  the  error 
due  to  the  difference  in  length  of  a  chord  and  its  arc. 

PROBLEM  26.  To  construct  an  Epicycloid  when  the  di- 
ameter of  the  generating  circle  and  the  diameter  of  the  director 
circle  are  given. 

The  epicycloid  and  hypocycloid  may  be  drawn  in  the  same 
manner  as  the  cycloid  if  arcs  of  circles  are  used  in  place  of  the 
horizontal  lines.  With  O  as  a  center  and  a  radius  of  f  inch 
describe  a  circle.  Draw  the  diameter  E  F  of  this  circle  and  pro- 
duce E  F  to  G  such  that  the  line  F  G  is  2f  inches  long.  With 
G  as  a  center  and  a  radius  of  2f  inches  describe  the  arc  A  B  of 
the  director  circle.  With  the  same  center  G,  draw  the  arc  P  Q 
which  will  be  the  path  of  the  center  of  the  generating  circle  as  it 
rolls  along  the  arc  A  fc,  Now  divide  the  generating  circle  into 


66  MECHANICAL  DRAWING. 

any  number  of  equal  parts  (twelve  for  instance)  and  through  the 
points  of  division  H,  I,  L,  M,  and  N,  draw  arcs  having  G  as  a 
center.  With  the  dividers  set  so  that  the  distance  between  the 
points  is  equal  to  the  chord  H  I,  mark  off  distances  on  the 
director  circle  A  F  B.  Through  these  points  of  division  R,  S, 
T,  U,  etc.,  draw  radii  intersecting  the  arc  P  Q  in  the  points  R',  S', 
T',  etc.,  and  with  these  points  as  centers  describe  arcs  of  circles 
as  in  Problem  25.  The  intersections  of  these  arcs  with  the  arcs 
already  drawn  through  the  points  H,  I,  L,  M,  etc.,  are  points  on 
the  curve.  Thus  the  intersection  of  the  circle  whose  center  is  R' 
with  the  arc  drawn  through  the  point  H  is  a  point  upon  the  curve. 
Also  the  arc  whose  center  is  S'  with  the  arc  drawn  through  the 
point  I  is  another  point  on  the  curve.  The  remaining  points  are 
found  by  repeating  this  process. 

PROBLEM  27.  To  draw  an  Hypocycloid  when  the  diam- 
eter of  the  generating  circle  and  the  radius  of  the  director  circle 
are  given. 

With  O  as  a  center  and  a  radius  of  4  inches  describe  the  arc 
E  F,  which  is  the  arc  of  the  director  circle.  Now  with  the  same 
center  and  a  radius  of  3^  inches,  describe  the  arc  A  B,  which  is  the 
line  of  centers  of  the  generating  circle  as  it  rolls  on  the  director 
circle.  With  O'  as  a  center  and  a  radius  of  |  inch  describe  the 
generating  circle.  As  before,  divide  the  generating  circle  into 
any  number  of  equal  parts  (12  for  instance)  and  with  these  points 
of  division  L,  M,  N,  O,  etc.,  draw  arcs  having  O  as  a  center. 
Upon  the  arc  E  F,  lay  off  distances  Q  R,  R  S,  S  T,  etc.,  equal  to 
the  chord  Q  L.  Draw  radii  from  the  points  R,  S,  T,  etc.,  to  the 
center  of  the  director  circle  O  and  describe  arcs  of  circles  having  a 
radius  equal  to  the  radius  of  the  generating  circle,  using  the 
points  G,  I,  J,  etc.,  as  centers.  As  in  Problem  26,  the  inter- 
sections of  the  arcs  are  the  points  on  the  curve.  By  repeating 
this  process,  the  right-hand  portion  of  the  curve  may  be  drawn. 

PROBLEM  28.  To  draw  the  Involute  of  a  circle  wh«n  the 
diameter  of  the  base  circle  is  known. 

With  point  O  as  a  center  and  a  radius  of  1  inch,  describe  the 
base  circle.  Now  divide  the  circle  into  any  number  of  equal  parts 
16  for  instance)  and  connect  the  points  of  division  with  the  cen- 


MECHANICAL  DRAWING.  67 

ter  of  the  circle  by  drawing  the  radii  O  C,  O  D,  O  E,  O  F,  etc., 
to  O  B.  At  the  point  D,  draw  a  light  pencil  line  perpendicular 
to  the  radius  O  D.  This  line  will  be  tangent  to  the  circle. 
Similarly  at  the  points  E,  F,  G,  H,  etc.,  draw  tangents  to  the 
circle.  Now  set  the  dividers  so  that  the  distance  between  the 
points  will  be  equal  to  the  chord  of  the  arc  C  D,  and  measure  this 
distance  from  D  akrag  the  tangent.  Beginning  with  the  point  E, 
measure  on  the  tangent  a  distance  equal  to  two  of  these  chords, 
from  the  point  F  measure  on  the  tangent  three  divisions,  and  from 
the  point  G  measure  a  distance  equal  to  four  divisions  on  the 
tangent  G  P.  Similarly,  measure  distances  on  the  remaining 
tangents,  each  time  adding  the  length  of  the  chord.  This  will 
give  the  points  Q,  R,  S  and  T.  Now  sketch  a  light  pencil  line 
through  the  points  L,  M,  N,  P,  etc.,  to  T.  This  curve  will  be  the 
involute  of  the  circle. 

Inking.  The  same  rules  are  to  be  observed  in  inking  PLATE 
VIII  as  were  followed  in  the  previous  plates,  that  is,  the  curves 
should  be  inked  in  a  full  line,  using  the  French  or  irregular  curve. 
All  arcs  and  lines  used  in  locating  the  points  on  one-half  of  the 
curve  should  be  inked  in  dotted  lines.  The  arcs  and  lines  used  in 
locating  the  points  of  the  other  half  of  the  curve  may  be  left  in 
pencil  in  Problems  25  and  26.  In  Problem  28,  all  construction 
lines  should  be  inked.  After  completing  the  problems  the  same 
lettering  should  be  done  on  this  plate  as  on  previous  plates. 


CIRCULAR  BLUE  PRINTING  MACHINE,  GENERAL  ELECTRIC  CO. 


MECHANICAL    DRAWING. 

PAKT  III. 


PROJECTIONS. 


ORTHOGRAPHIC   PROJECTION. 

Orthographic  Projection  is  the  art  of  representing  objects  of 
three  dimensions  by  views  on  two  planes  at  right  angles  to  each 
other,  in  such  a  way  that  the  forms  and  positions  may  be  completely 
determined.  The  two  planes  are  called  planes  of  projection  or 
co-ordinate  planes,  one  being  vertical  and  the  other  horizontal,  as 
shown  in  Fig.  1.  These  planes  are  sometimes  designated  V  and  H 
respectively.  The  intersection  of  V  and  H  is  known  as  the  ground 
line  G  L. 

The  view  or  projection  of  the  figure  on  the  plane  gives  the 
same  appearance  to  the  eye  placed  in  a  certain  position  that  the 
object  itself  does.  This  position 
of  the  eye  is  at  an  infinite  dist- 
ance from  the  plane  so  that  the 
rays  from  it  to  points  of  a  limited 
object  are  all  perpendicular  to  the 
plane.  Evidently  then  the  view  of 
a  point  of  the  object  is  on  the  plane 
and  in  the  ray  through  the  point  Fig.  1. 

and  the  eye  or  where  this  perpendicular  to  the  plane  pierces  it. 

Let  a,  Fig.  1,  be  a  point  in  space,  draw  a  perpendicular  from  a 
to  V.  Where  this  line  strikes  the  vertical  plane,  the  projection  of  a 
is  found,  namely  at  «v.  Then  drop  a  perpendicular  from  a  to  the 
horizontal  plane  striking  it  at  «h,  which  is  the  horizontal  projection 
of  the  point.  Drop  a  perpendicular  from  #v  to  H;  this  will 
intersect  G  L  at  o  and  be  parallel  and  equal  to  the  line  a  0h.  In 
the  same  way  draw  a  perpendicular  from  «h  to  V,  this  also  will 
intersect  G  L  at  o  and  will  be  parallel  and  equal  to  a  av.  In  other 
words,  the  perpendicular  to  G  L  from  the  projection  of  a  point  on 
either  plane  equals  the  distance  of  the  point  from  the  other  plane. 
B  in  Fig.  1,  shows  a  line  in  space.  Bv  is  its  V  projection,  and  Bh 

Copyright,  1908,  by  American  School  of  Correspondence. 


MECHANICAL   DRAWING 


its  H  projection,  these  being  determined  by  finding  views  of  points 
at  its  ends  and  connecting  the  points. 

Instead  of  horizontal  projection  and  vertical  projection,  the 
terms  plan  and  elevation  are  commonly  used. 

Suppose  a  cube,  one  inch  on  a  side,  to  be  placed  as  in  Fig.  2, 
with  the  top  face  horizontal  and  the  front  face  parallel  to  the 
vertical  plane.  Then  the  plan  will  be  a  one-inch  square,  and  the 
elevation  also  'a  one-inch  square.  In  general  the  plan  is  a  repre- 
sentation of  the  top  of  the  object,  and  the  elevation  a  view  of  the 
front.  The  plan  then  is  a  top  view,  and  the  elevation  a  front  view. 


V 

r  5 

ev 

|H 

. 

M 

<T 



3H 

H 

Fig.  2. 


Fig.  3. 


Thus  far  the  two  planes  have  been  represented  at  right  angles 
to  each  other,  as  they  are  in  space.  In  order  that  they  may  be 
shown  more  simply  and  on  the  one  plane  of  the  paper,  H  is 
revolved  about  G  L  as  an  axis  until  it  lies  in  the  same  plane  as  V 
as  shown  in  Fig.  2.  The  lines  lh  O  and  2h  N,  being  perpendicular 
to  G  L,  are  in  the  same  straight  line  as  5V  O  and  6V  N,  which  also 
are  perpendicular  to  G  L.  That  is — two  views  of  a  point  are 
always  in  a  line  perpendicular  to  O  Z.  From  this  it  is  evident 
that  the  plan  must  be  vertically  below  the  elevation,  point  for  point. 
Now  looking  directly  at  the  two  planes  in  the  revolved  position,  we 


MECHANICAL   DRAWING  71 

get  a  true  orthographic  projection  of  the  cube  as  shown  in  Fig.  3. 
All  points  on  an  object  at  the  same  height  must  appear  in 
elevation  at  the  same  distance  above  the  ground  line.  If  numbers 
1,  2,  3,  and  4  on  the  plan,  Fig.  3,  indicate  the  top  corners  of  the 
cube,  then  these  four  points,  being  at  the  same  height,  must  be 

4V          3V 
shown  in  elevation  at  the  same  height  and  at  the  top.    -and -^ 

The  top  of  the  cube,  1,  2, 3, 4,  is  shown  in  elevation  as  the  straight  line 

4V    3V 

—  -' _-  .   This  illustrates  the  fact  that  if  a  surface  is  perpendicular 

to  either  plane  or. projection,  its  projection  on  that  plane  is  simply 
a  line;  a,  straight  line  if  the  surface  is  plane,  a  curved  line  if  the 
surface  is  curved.  From  the  same  figure  it  is  seen  that  the  top 
edge  of  the  cube,  1  4,  has  for  its  projection  on  the  vertical  plane 

A  v 

the  point  -y^  the  principle  of  which   is  stated  in  this  way:  If  a 


straight  line  is  perpendicular  to  either  V or  H,  its  projection  on 
that  plane  is  a  point,  and  on  the  other  plane  is  a  line  equal  in 
length  to  the  line  itself,  and  perpendicular  to  the  ground  line.^ 

Fig.  4  is  given  as  an  exercise  to  help  to  show  clearly  the  idea 
of  plan  and  elevation. 

A  =  a  point  B"  above  H,  and  A"  in  front  of  V. 

B  =  square  prism  resting  on  H,  two  of  its  faces  parallel  to  V, 

C  =  circular  disc  in  space  parallel  to  V. 

D  =  triangular  card  in  space  parallel  to  V. 

E  =  cone  resting  on  its  base  on  H. 

F  =  cylinder  perpendicular  to  V,  and  with  one  end  resting  against  V. 

G  =  line  perpendicular  to  H. 

H  =  triangular  pyramid  above  H,  with  its  base  resting  against  V. 


MECHANICAL  DRAWING. 


Suppose  in  Fig.  5,  that  it  is  desired  to  construct  the  pro- 
jections of  a  prism  1£  in.  square,  and  2  in.  long,  standing  on  one 
end  on  the  horizontal  plane,  two  of  its  faces  being  parallel  to  the 
vertical  plane.  In  the  first  place,  as  the  top  end  of  the  prism  is  a 
square,  the  top  view  or  plan  will  be  a  square  of  the  same  size, 
that  is,  1|  in.  Then  since  the  prism  is  placed  parallel  to  and  in 
front  of  the  vertical  plane  the  plan,  1£  in.  square,  will  have  two 
edges  parallel  to  the  ground  line.  As  the  front  face  of  the  prisn- 


T 

1 

ELEVATION 

I 

! 

OR 

J 
OJ 

ELEVATION 

FRONT      VIEW 

1 
1 

1 
1 

— if — 

PLAN 

OR 
TOP     VIEW 


Pig.  6. 


is  parallel  to  the  vertical  plane  its  projection  on  V  will  be  a  rect- 
angle, equal  in  length  and  width  to  the  length  and  width  respec- 
tively of  the  prism,  and  as  the  prism  stands  with  its  base  on  H, 
the  elevation,  showing  height  above  H,  must  have  its  base  on  the 
ground  line.  Observe  carefully  that  points  in  elevation  are  verti- 
cally over  corresponding  points  in  plan. 

The  second  drawing  in  Fig.  6  represents  a  prism  of  the  same 
size  lying  on  one  side  on  the  horizontal  plane,  and  with  the  en'te 
parallel  to  V. 

The  principles  which  have  been  used  thus  far  may  be  stated 
as  follows,  — 


MECHANICAL  DRAWING. 


73 


1.  If  a  line  or  point  is  on  either  plane,  its  other  projection 
must  be  in  the  ground  line. 

2.  Height  above  H  is  shown  in  elevation  as  height  above 
the  ground  line,  and  distance  in  front  of  the  vertical  plane  is  shown 
in  plan  as  distance  from  the  ground  line. 

3.  If  a  line  is  parallel  to  either  plane,  its  actual  length  is 
shown  on  that  plane,  and    its  other  projection  is  parallel  to  the 
ground  line.     A  line  oblique  to  either  plane  has  its  projection  on 
that  plane  shorter  than  the  line  itself,  and  its    other  projection 
oblique  to  the  ground  line.    No  projection  can  be  longer  than  the 
line  itself. 

4.  A  plane  surface  if  parallel  to  either  plane,  is  shown  OD 


Fig.  <k 


Wg. 


that  plane  in  its  true  size  and  shape ;  if  oblique  it  is  shown 
smaller  than  the  true  size,  and  if  perpendicular  it  is  shown  as  a 
straight  line.  Lines  parallel  in  space  must  have  their  V  projec- 
tions parallel  to  each  other  and  also  their  H  projections. 

If  two  lines  intersect,  their  projections  must  cross,  since  the 
point  of  intersection  of  the  lines  is  a  point  on.  both  lines,  and 
therefore  the  projections  of  this  point  must  be  on  the  projections 
of  both  lines,  or  at  their  intersection.  In  order  that  intersecting 
lines  may  be  represented,  the  vertical  projections  must  intersect 
in  a  point  vertically  above  the  intersection  of  the  horizontal  pro 


74 


MECHANICAL  DRAWING. 


jections.  Thus  Fig.  6  represents  two  lines  which  do  intersect  as 
O  crosses  D"at  a  point  vertically  above  the  intersection  of  Ch  and 
Dfc.  In  Fig.  7,  however,  the  lines  do  not  intersect  since  the  inten- 
sections  of  their  projections  do  not  lie  in  the  same  vertical  line. 

In  Fig.  8  is  given  the  plan  and  elevation  of  a  square  pyramid 
standing  on  the  horizontal  plane.  The  height  of  the  pyramid  is 
the  distance  A  B.  The  slanting  edges  of  the  pyramid,  A  C,  A  D, 
etc.,  must  be  all  of  the  same  length, 'since  A  is  directly  above  the 
center  of  the  base.  What  this  length 
is,  however,  does  not  appear  in  either 
projection,  as  these  edges  are  not 
parallel  to  either  V  or  H. 

Suppose  that  the  pyramid  be 
turned  around  into  the  dotted  posi- 
tion 0,  D,  E,  F,  where  the  horizontal 
projections  of  two  of  the  slanting 
edges,  A  C,  and  A  E,  are  parallel  to 
the  ground  line.  These  two  edges, 
having  their  horizontal  projections 
parallel  to  the  ground  line,  are  now 
parallel  to  V,  and  therefore  their  new 
vertical  projections  will  show  their 
true  lengths.  The  base  of  the  pyra- 
mid is  still  on  H,  and  therefore  is 
projected  on  V  in  the  ground  line. 
The  apex  is  in  the  same  place  as  be- 
fore, hence  the  vertical  projection  of 
the  pyramid  in  its  new  position  is  shown  by  the  dotted  lines.  The 
vertical  projection  A  C^is  the  true  length  of  edge  A  C.  Now  if 
we  wish  to  find  simply  the  true  length  of  A  C,  it  is  unnecessary  to 
turn  the  whole  pyramid  around,  as  the  one  line  A  C  will  be  sufficient. 
The  principle  of  finding  the  true  length  of  lines  is  this,  anu 
can  be  applied  to  any  case  :  Swing  one  projection  of  the  line  par- 
allel to  the  ground  line,  using  one  end  as  center.  On  the  other 
projection  the  moving  end  remains  at  the  same  distance  from  the 
ground  line,  and  of  course  vertically  above  or  below  the  same  end 
in  its  parallel  position.  This  new  projection  of  the  line  shows  its 
true  length.  See  the  three  Figures  at  the  top  of  page  9. 


MECHANICAL  DRAWING. 


75 


Third  plane  of  projection  or  profile  plane.       A  plane  peipen- 
dicular  to  both  co-ordinate  planes,  and  lieuce'to  the  ground  line,  is 


called  a  profile  plane.  This  plane  is  vertical  in  position,  and  may 
be  used  as  a  plane  of  projection.  A  projection  on  the  profile  plane 
is  called  a  profile  view,  or  end  view,  or  sometimes  edge  view,  and 
is  often  required  in  machine  or  other  drawing  when  the  plan  and 
elevation  do  not  sufficiently  give  the  shape  and  dimensions. 

A  projection  on  this  plane  is  found  in  the  same  way  as  on  the 


Fig.  9. 


V  plane,  that   is,  by  perpendiculars    drawn  from  points  on  the 
object. 

Since,    however,  the    profile    plane   is   perpendicular  to  the 
ground  line,  it  will  b«  seen  from  the  front  and  top  simply  as  a 


76 


MECHANICAL  DRAWING. 


straight  line ;  in  order  that  the  size  and  shape  of  the  profile  view 
may  be  shown,  the  profile  plane  is  revolved  into  V  using  its  inter 
section  with  the  vertical  plane  as  the  axis. 

Given  in  Fig.  9,  the  line  A  B  by  its  two  projections  A*  B°  and 
Ah  BA,  and  given  also  the  profile  plane.  Now  by  projecting  the 
line  on  the  profile  by  perpendiculars,  the  points  A,*  B,*  and  B,fc  A,* 
are  found.  Revolving  the  profile  plane  like  a  door  on  its  hinges,  alJ 
points  in  the  plane  will  move  in  horizontal  circles,  so  the  horizontal 
projections  A,71  and  B,fc  will  move  in  arcs  of  circles  with  O  as  center 
to  the  ground  line,  and  the  vertical  projections  B,"  and  A,*  will  move 
in  lines  parallel  to  the  ground  line  to  positions  directly  above  the 
revolved  points  in  the  ground  line,  giving  the  profile  view  of  the 
line  Ap  Bp.  Heights,  it  will  be  seen,  are  the  same  in  profile  view 

as  in  elevation.  By  referring  to 
the  rectangular  prism  in  the  same 
figure,  we  see  that  the  elevation 
gives  vertical  dimensions  and  those 
parallel  to  V,  while  the  end  view 
shows  vertical  dimensions  and 
those  perpendicular  to  V.  The 
profile  view  of  any  object  may  be 
found  as  shown  for  the  line  A  B 
by  taking  one  point  at  a  time. 

In  Fig.  10  there  is  repre- 
sented a  rectangular  prism  or 
block,  whose  length  is  twice  the 
width.  The  elevation  shows  its 
height.  As  the  prism  is  placed  at 
an  angle,  three  of  the  vertical  edges  will  be  visible,  the  fourth 
one  being  invisible. 

In  mechanical  drawing  lines  or  edges  which  are  invisible  are 
drawn  dotted.  The  edges  which  in  projection  form  a  part  of  the 
outline  or  contour  of  the  figure  must  always  be  visible,  hence 
always  full  lines.  The  plan  shows  what  lines  are  visible  in  eleva- 
tion, i'.nd  the  elevation  determines  what  are  visible  in  plan.  In 
Fig.  1 0,  the  plan  shows  that  the  dotted  edge  A  B  is  the  back  edge, 
and  in  Fig.  11,  the  dotted  edge  C  D  is  found,  by  looking  at  the 
elevation,  to  oe  the  lower  edge  of  the  triangular  prism.  In  general, 


Fig.  10. 


MECHANICAL  DRAWING. 


77 


if  in  elevation  an  edge  projected  within  the  figure  is  a  back  edge, 
it  must  ^be  dotted,  and  in  plan  if  an  edge  projected  within  the 
outline  is  a  lower  edge  it  is  dotted. 

Fig.  12  is  a  circular  cylinder  with  the  length  vertical  and 


Fig.  11. 

with  a  hole  part  way  through  as  shown  in  elevation.  Fig.  13  is 
plan,  elevation  and  end  view  of  a  triangular  prism  with  a  square 
hole  from  end  to  end.  The  plan  and  elevation  alone  would  be 
insufficient  to  determine  positively  the  shape  of  the  hole,  but  the 
end  view  shows  at  a  glance  that  it  is  square. 

In  Fig.  14  is  shown  plan  and  elevation  of  the  frustum  of  a 
square  pyramid,  placed  with  its  base  011  the  horizontal  plane.  If  the 
frustum  is  turned  through  30°,  as  shown  in  the  plan  of  Fig,  15, 
the  top  view  or  plan  must  still  be  the  same  shape  and  size,  and  as 
the  frustum  has  not  been  raised  or  lowered,  the  heights  of  all 
points  must  appear  the  same  in  elevation  as  before  in  Fig.  14. 
The  elevation  is  easily  found  by  projecting  points  up  from  the 
plan,  and  projecting  the  height  of  the  top  horizontally  across  from 
the  first  elevation,  because  the  height  does  not  change. 

The  same  principle  is  further  illustrated  in  Figs.  16  and  17. 
The  elevation  of  Fig.  16  shows  a  square  prism  resting  on  one  edge, 
and  raised  up  at  an  angle  of  30°  on  the  right-hand  side.  The 


78 


MECHANICAL  DRAWING. 


plan  gives  the  width  or  thickness,  $  in.     Notice  that  the  length  ol 
the  plan   is  greater  than  2  in.   and   that   varying  the  angle  at 


Fig.  12. 


Fig.  13. 


which  the  pripni  is  slanted  would  change  the  length  of  the  plan. 
Now  if  the  prism  be  turned  around  through  any  angle  with  the 
vertical  plane,  the  lower  edge  still  being  on  H,  and  the  inclination 


Fig.  14. 


Fig.  15. 


of  30°  with  H  remaining  the  same,  the  plan  must  remain  the  same 
size  and  shape. 

If  the  angle  through  which  the  prism  be  turned  is  45°,  we 


MECHANICAL  DRAWING. 


79 


have  the  second  plan,  exactly  the  same  shape  and  size  as  the  first 
The  elevation  is  found  by  projecting  the  corners  of  the  prism  vei> 


Fig.  16. 


tically  up  to  the  heights  of  the  same  points  in  the  first  elevation. 
All  the  other  points  are  found  in  the  same  way  as  point  No.  1. 


Fig.  17. 


Three  positions  of  a  rectangular  prism  are  shown  in  Fig.  17. 
In  the  first  view,  the  prism  stands  on  its  hase.  its  axis  therefore 


80 


MECHANICAL  DRAWING. 


is  parallel  to  the  vertical 'plane.  In  the  second  position,  the. axis  ie 
still  parallel  to  V  and  one  corner  of  the  base  is  on  the  horizontal 
plane.  The  prism  has  been  turned  as  if  on  the  line  I*  lr  as  an 
axis,  so  that  the  inclination  of  all  the  faces  of  the  prism  to  the 
vertical  plane  remains  the  same  as  before.  That  is,  if  in  the  first 
figure  the  side  A  B  C  D  makes  an  angle  of  30°  with  the  vertical, 
the  same  side  in  the  second  position  still  makes  30°  with  the  ver- 


.  16. 


tical  plane.  Hence  the  elevation  of  No.  2  is  the  same  shape  and  size 
us  in  the  first  case.  The  plan  is  found  by  projecting  the  corner? 
down  from  the  elevation  to  meet  horizontal  lines  projected  across 
from  the  corresponding  points  in  the  first  plan.  The  third  posi- 
tion shows  the  prism  with  all  its  faces  and  edges  making  the  same 
angles  with  the  horizontal  as  in  the  second  position,  but  with  the 
plan  at  a  different  angle  with  the  ground  line.  The  plan  then  is 
the  same  shape  and  size  as  in  No.  2,  and  the  elevation  is  found  by 
projecting  up  to  the  same  heights  as  shown  in  the  proceeding 
elevation.  This  principle  may  be  applied  to  any  solid,  whether 
bounded  by  plane  surfaces  or  curved. 

This  principle  as  far  as  it  relates  to  heights,  is  the  same  that 
was  used  for  profile  views.  An  end  view  is  sometimes  necessary 
before  the  plan  or  elevation  of  an  object  can  be  drawn.  Suppose 
that  in  Fig.  18  we  wish  to  draw  the  plan  and  elevation  of  a  tri- 
angular prism  3"  long,  the  end  of  which  is  an  equilateral  triangle 


MECHANICAL  DRAWING. 


si 


U"  on  each  side.     The  prism  is  lying  on  one  of  its  three  faces  on 
H,  and  inclined  toward  the  vertical  plane  at  an  angle  of  30°.    We 

are  able  to  draw  the  plan  at 
once,  because  the  width  will  be 
1|  inches, 'and  the  top  edge  will 
be  projected  half  way  between 
the  other  two.  The  length  of 
the  prism  will  also  be  shown. 
Before  we  can  draw  the  elevation, 
we  must  find  the  height  of  the 
top  edge.  This  height,  however, 
must  be  equal  to  the  altitude  of 
the  triangle  forming  the  end  of 

Fig.  19.  the  prism.    All  that  is  necessary, 

then,  is  to  construct  an  equilat- 
eral triangle  li"  on  each  side,  and  measure  its  altitude. 

A  very  convenient  way  to  do  this  is  shown  in  the  figure  by 
laying  one  end  of  the  prism  down  on  H.  A  similar  construction 
is  shown  in  Fig.  19,  but  with  one  face  of  the  prism  on  V  instead 
of  on  H. 

In  all  the  work  thus  far  the  plan  has  been  drawn  below  and 
the  elevation  above.  This  order  is  sometimes  inverted  and  the 
plan  put  above  the  elevation,  but  the  plan  still  remains  a  top  view 
no  matter  where  placed,  so  that  after  some  practice.it  makes  but 
little  difference  to  the  draughtsman  which  method  is  employed. 

SHADE  LINES. 

It  is  often  the  case  in  machine  drawing  that  certain  lines  or 
edges  are  made  heavier  than  others.  These  heavy  lines  are  called 
shade  lines,  and  are  used  to  improve  the  appearance  *of  the  draw- 
ing, and  also  to  make  clearer  in  some  cases  the  shape  of  the 
object.  The  shade  lines  are  not  put  on  at  random,  but  according 
to  some  system.  Several  systems  are  in  use,  but  only  that  one 
which  seems  most  consistent  will  be  described.  The  shade  lines 
are  lines  or  edges  separating  light  faces  from  dark  ones,  assuming 
the  light  always  to  come  in  a  direction  parallel  to  the  dotted 
diagonal  of  the  cube  shown  in  Fig.  20.  The  direction  of  the 
light,  then,  may  be  represented  on  H  by  a  line  at  45°  running 


SI' 


MECHANICAL  DRAWING. 


backward  to  the  right  and  on  V  by  a  45°  line  sloping  downward 
and  to  the  right.  Considering  the  cube  in  Fig.  20,  if  the  light 
comes  in  the  direction  indicated,  it  is  evident  that  the  front,  left- 
hand  side  and  top  will  be  lightj  and  the  bottom,  back  and  right- 
hand  side  dark.  On  the  plan,  then,  the  shade  lines  will  be  the 
back  edge  1  2  and  the  right-hand  edge  2  3,  because  these  edges 
are  between  light  faces  and  dark  ones.  On  the  elevation,  since 
the  front  is  light,  and  the  right-hand  side  and  bottom  dark,  the  edges 
3  7  and  8  7  are  shaded.  As  the  direction  of  the  light  is  represented 
on  the  plan  by  45°  lines  and  on  the  elevation  also  by  45°  lines, 


1 

2 
\ 

\ 

\ 
\ 

\ 
\ 

\ 

\ 

8 

1 

5                                " 

5                           / 
/ 

X 

^ 

X 

/ 

X 

4 

S                             3 

8          /I             / 

Fig.  20. 

we  may  use  the  45°  triangle  with  the  T-square  to  determine 
the  light  and  dark  surfaces,  and  hence  the  shade  lines.  If 
the  object  stands  on  the  horizontal  plane,  the  45°  triangle  is  used 
on  the  plan,  as  shown  in  Fig.  21,  but  if  the  length  is  perpen- 
dicular to  the  vertical  plane,  the  45°  triangle  is  used  on  the  eleva- 
tion, as  shown  in  Fig.  22.  This  is  another  way  of  saying  that  the 
45°  triangle  is  used  on  that  projection  of  the  object  which  shows 
the  end.  By  applying  the  triangle  in  this  way  we  determine  the 
light  and  dark  surfaces,  and  then  put  the  shade  lines  between 
them.  Dotted  lines,  however,  are  never  shaded,  so  if  a  line 
which  is  between  a  light  and  a  dark  surface  is  invisible  it  is  not 


MECHANICAL    DRAWING. 


shaded.     In  Fig.  21  the  plan  shows  the  end  of  the  solid,  hence  the 
45°  triangle  is  used  in  the  direction  indicated  by  the  arrows. 

This  shows  that  the  light  strikes  the  left-hand  face,  but  not 
the  back  or  the  right-hand.     The  top  is  known  to  be  light  with- 


Fig.  21. 


Fig.  22. 


out  the  triangle,  as  the  light  comes  downward,  so  the  shade  edges 
on  the  plan  are  the  back  and  right-hand.  On  the  elevation  two 
faces  of  the  prism  are  visible ;  one  is  light,  the  other  dark,  hence 
the  edge  between  is  shaded.  The  left-hand  edge,  being  between 
a  light  face  and  a  dark  one  is  a  shade  line.  The  right-hand  face 
is  dark,  the  top  of  the  prism  is  light,  hence  the  upper  edge  of  this 
face  is  a  shade  line.  The  right-hand  edge  is  not  shaded,  because 
by  referring  to  the  plan,  it  is  seen  to  be  between  two  dark 
surfaces.  In  shading  a  cylinder  or  a  cone  the  same  rule  is  fol 
lowed,  the  only  difference  being  that  as  the  surface  is  curved,  the- 
light  is  tangent,  so  an  element  instead  of  an  edge  marks  the 
separation  of  the  dark  from  the  light,  and  is  not  shaded.  The 
elements  of  a  cylinder  or  cone  should  never  be  shaded,  but  the 
bases  may.  In  Fig.  23,  Nos.  3  and  4,  the  student  should  carefully 
notice  the  difference  between  the  shading  of  the  cone  and  cylinder. 


84 


MECHANICAL     DRAWING. 


If  in  No.  4  the  cone  were  inverted,  the  opposite  half  of  the  base 
would  be  shaded,  for  then  the  base  would  be  light,  whereas  it  is 
now  dark.  In  Nos.  7  and  8  the  shade  lines  of  a  cylinder  and  a 
circular  hole  are  contrasted. 

In  No.  7  it  is  clear  that  the  light  would  strike  inside  on  the 
further  side  of  the  hole,  commencing  half  way  where  the  45°  lines 

1234 


678 

Fig.  23. 

are  tangent.  The  other  half  of  the  inner  surface  would  be  dark, 
hence  the  position  of  the  shade  line.  The  shade  line  then  enables 
us  to  tell  at  a  glance  whether  a  circle  represents  a  hub  or  boss,  or 
depression  or  hole.  Fig.  24  represents  plan,  elevation  and  profile 
view  of  a  square  prism.  Here  as  before,  the  view  showing  the 
end  is  the  one  used  to  determine  the  light  and  dark  surfaces,  and 
then  the  shade  lines  put  in  accordingly. 


MECHANICAL  DRAWING. 


^  In  putting  on  the  shade  lines,  the  extra  width  of  line  is  put 
inside  the  figure,  not  outside.  In  shading  circles,  the  shade  line 
is  made  of  varying  width,  as  shown  in  the  figures.  The  method 
of  obtaining  this  effect  by  the  compass  is  to  keep  the  same  radius, 
but  to  change  the  center  slightly  in  a  direction  parallel  to  the  rays 
of  light,  as  shown  at  A  and  B  in  No.  2  of  Fig.  24. 

No.  2. 


Fig.  24. 

INTERSECTION  AND  DEVELOPHENT. 

If  one  surface  meets  another  at  some  angle,  an  intersection  is 
produced.  Either  surface  may  be  plane,  or  curved.  If  both  are 
plane,  the  intersection  is  a  straight  line ;  if  one  is  curved,  the 
intersection  is  a  curve,  except  in  a  few  special  cases  ;  and  if  both 
are  curved,  the  intersection  is  usually  curved. 

In  the  latter  case,  the  entire  curve  does  not  always  lie  in  the 
same  planes.  If  all  points  of  any  curve  lie  in  the  same  plane,  it 
is  called  a  plane  curve.  A  plane  intersecting  a  curved  surface 
must  ahvays  give  either  a  plane  curve  or  a  straight  line. 

In  Fig.  25  a  square  pyramid  is  cut  by  a  plane  A  parallel  to  the 
horizontal.  This  plane  cuts  from  the  pyramid  a  four-sided  figure, 
the  four  corners  of  which  will  be  the  points  where  A  cuts  the  four 
slanting  edges  of  the  solid.  The  plane  intersects  edge  o  I  at  point  4^ 
in  elevation.  This  point  must  be  found  in  plan  vertically  below  on 


MECHANICAL     DRAWING. 


the  horizontal  projection  of  line  o  b,  that  is,  at  point  4^.  Edge 
0  e  is  directly  in  front  of  o  5,  so  is  shown  in  elevation  as  the  same 
line,  and  plane  A  intersects  o  e  at  point  1"  in  elevation,  found  in 
plan  at  1A  Points  3  and  2  are  obtained  in  the  same  way.  The 
intersection  is  shown  in  plan  as  the  square  1234,  which  is  also 
its  true  size  as  it  is  parallel  to  the  horizontal  plane.  In  a 

similar  way  the  sections  are  found 
in  Figs.  26  and  27.  It  will  be 
seen  that  in  these  three  cases 
where  the  planes  are  parallel  to 
the  bases,  the  sections  are  of  the 
same  shape  as  the  bases,  and  have 
their  sides  parallel  to  the  edges  of 
the  bases. 

It  is  an  invariable  rule  that 
when  such  a  solid  is  cut  by  a  plane 
parallel  to  its  base,  the  section  is 
a  figure  of  the  same  shape  as  the 
base.  If  then  in  Fig.  28  a  right 
cone  is  intersected  by  a  plane 
parallel  to  the  base  the  section 
must  be  a  circle,  the  center  of 
which  in  plan  coincides  with  the  apex..  The  radius  must 
equal  o  d. 

In  Figs.  ii9  and  30  the  cutting  plane  is  not  parallel  to  the  base, 
hence  the  intersection  will  not  be  of  the  same  shape  as  the  base. 
The  sections  are  found,  however,  in  exactly  the  same  manner  as 
in  the  previous  figures,  by  projecting  the  points  where  the  plane 
intersects  the  edges  in  elevation  on  to  the  other  view  of  the  same 
line. 

INTERSECTION  OF  PLANES  WITH  CONES  OR  CYLINDERS. 

Sections  cut  by  a  plane  from  a  cone  have  already  been  de- 
fined as  conic  sections.  These  sections  may  be  either  of  the  fol- 
lowing: two  straight  lines,  circle,  ellipse,  parabola,  hyperbola. 
All  except  the  parabola  and  hyperbola  may  also  be  cut  from  a 
cylinder. 

Methods  have  previously  been   given  tor  constructing  the 


Fig.  25. 


MECHANICAL     DRAWING. 


87 


Fig.  26. 


Fig.  27. 


'    \\ 


Fig.  28. 


Fig.  29. 


Fig.  30. 


ss 


MECHANICAL     DRAWING. 


ellipse,  parabola  and  hyperbola  without  projections;  it  will  now 
be  shown  that  they  may  be  obtained  as  actual  intersections. 

In  Fig.    31   the    plane    cuts  the  cone  obliquely.      To    find 
points   on  the  curve  in  plan  take  a  series  of  horizontal   planes 


Fig.  31 

x  y  z  etc.,  between  points  <?»  and  ifr.  One  of  these  planes,  as  u.\ 
should  be  taken  through  the  center  of  c  d.  The  points  c  and  d 
must  be  points  on  the  curve,  since  the  plane  cuts  the  two  contoui 
elements  at  these  points.  The  horizontal  projections  of  the  contour 
elements  will  be  found  in  a  horizontal  line  passing  through  the  center 
of  the  base ;  hence  the  horizontal  projection  of  c  and  d  will  be 
found  on  this  center  line,  and  will  be  the  extreme  ends  of  the 
curve.  Contour  elements  are  those  forming  the  outline. 


MECHANICAL    DRAWING. 


The  plane  x  cuts  the  surface  of  the  cone  in  a,  circle,  as  it  is 
parallel  to  the  b.ise,  and  the  diameter  of  the  circle  is  the  distance 
between  the  points  where  x  crosses  the  two  contour  elements. 
This  circle,  lettered  x  on  the  plan,  has  its  center  at  the  horizontal 
projection  of  the  apex.  The  circle  x  and  the  curve  cut  by  the  plane 
are  both  on  the  surface  of  the  cone,  and  their  vertical  projec- 
tions intersect  at  the  point  2.  Also  the  circle  x  and  the  curve 
must  cross  twice,  once  on  the  front  of  the  cone  and  once  on  the 
back.  Point  2  then  represents  two  points  which  are  shown  in 
plan  directly  beneath  on  the  circle  x,  and  are  points  on  the  re- 
quired intersection.  Planes  y  and  z,  and  as  many  more  as  may 
be  necessary  to  determine  the  curve  accurately,  are  used  in  the 
same  way.  The  curve  found  is  an  ellipse.  The  student  will 
readily  see  that  the  true  size  of  this  ellipse  is  not  shown  in  the 
plan,  for  the  plane  containing  the  curve  is  not  parallel  to  the 
horizontal. 

In  order  to  find  the  actual  size  of  the  ellipse,  it  is  necessary 
to  place  its  plane  in  a  position  parallel  either  to  the  vertical  or  to 
the  horizontal.  The  actual  length  of  the  long  diameter  of  the 
ellipse  must  be  shown  in  elevation,  <?»  dv,  because  the  line  is 
parallel  to  the  vertical  plane.  The  plane  of  the  ellipse  then  may 
be  revolved  about  <?»  d°  as  an  axis  until  it  becomes  parallel  to  V, 
when  its  true  size  will  be  shown.  For  the  sake  of  clearness  of 
construction,  c*>  d°  is  imagined  moved  over  to  the  position  e?  d\ 
parallel  to  c®  d*>.  The  lines  1 — 1,  2 — 2,  3  —  3  on  the  plan  show  the 
true  width  of  the  ellipse,  as  these  lines  are  parallel  to  H,  but  are 
projected  closer  together  than  their  actual  distances.  In  elevation 
these  lines  are  shown  as  the  points  1,  2,  3.  at  their  true  distance 
apart.  Hence  if  the  ellipse  is  revolved  aro'inu.  its  axis  c°  f?c,  the 
distances  1 — 1,  2 — 2,  3 — 3  will  appear  perpendicular  to  c»tZ",  and 
the  true  size  of  the  figure  be  shown.  This  construction  is  made  on 
the  left,  where  1'— 1',  2'— 2'  and  3'— 3'  are  equal  in  length  to  1—1, 
2 — 2,  3 — 3  on  the  plan. 

In  Fig.  32  a  plane  cuts  a  cylinder  obliquely.  This  is  a 
simpler  case,  as  the  horizontal  projection  of  .the  curve  coincides 
with  the  base  of  the  cylinder.  To  obtain  the  true  size  of  the 
section,  which  is  an  ellipse,  any  number  of  points  are  assumed  on 
the  plan  and  projected  up  on  the  cutting  plane,  at  1,  2,  3,  etc. 


90 


MECHANICAL     DRAWING. 


The  lines  drawn  through   these  points  perpendicular  to  1  7  are 
made  equal  in  length  to  the  corresponding  distances  2' — 2',  3' — 3' 
etc.,  on  the  plan,  because  2' — 2'  is  the  true  width  of  curve  at  2. 
If  a  cone  is  intersected  by  a  plane  which  is  parallel  to  only 

one  of  the  elements,  as  in 
Fig.  33,  the  lesulting  curve 
is  the  parabola,  the  construc- 
tion of  which  is  exactly  simi- 
lar to  that  for  the  ellipse  as 
given  in  Fig.  31.  If  the 
intersecting  plane  is  parallel 
to  more  than  one  element,  or 
is  parallel  to  the  axis  of  the 
cone,  a  hyperbola  is  produced. 
In  Fig.  34,  the  vertical 
plane  A  is  parallel  to  the  axis 
of  the  cone.  In  this  instance 
the  curve  when  found  will 
appear  in  its  true  size,  as 
plane  A  is  parallel  to  the 
vertical.  Observe  that  the 
highest  point  of  the  curve  is 
found  by  drawing  the  circle 
X  on  the  plan  tangent  to  the 
given  plane.  One  of  the 
points  where  this  circle  crosses 
the  diameter  is  projected  up 
to  the  contour  element  of  the 
cone,  and  the  horizontal  plane  X  drawn.  Intermediate  planes 
Y,  Z,  etc.,  are  chosen,  and  corresponding  circles  drawn  in  plan. 
The  points  where  these  circles  are  crossed  by  the  plane  A  are 
points  on  the  curve,  and  these  points  are  projected  up  to  the 
elevation  on  the  planes  Y,  Z,  etc. 

DEVELOPflENTS. 

The  development  of  a  surface  is  the  true  size  and  shape  ot 
the  surface  extended  or  spread  out  on  a  plane.  If  the  surface  to 
be  developed  is  of  such  a  character  that  it  may  be  flattened  out 


MECHANICAL     DRAWING. 


91 


without  tearing  or  folding,  we  obtain  an  exact  development,  as  in 
case  of  a  cone  or  cylinder,  prism  or  pyramid.  If  this  cannot  be 
done,  as  with  the  sphere,  the  development  is  only  approximate. 

In  order  to  find  the  development  of  the  rectangular  prism  in 
Fig  35,  the  back  face,  1  2  7  6,  is  supposed  to  be  placed  in  contact 


Fig.  33. 

with  some  plane,  then  the  prism  turned  on  the  edge  2  7  until  the 
side  2  3  8  7  is  in  contact  with  the  same  plane,  then  this  continued 
until  all  four  faces  have  been  placed  on  the  same  plane.  The 
rectangles  1  4  3  2  and  6  7  8  5  are  for  the  top  and  bottom  respec- 
tively. The  development  then  is  the  exact  size  and  shape  of  a 
covering  for  the  prism.  If  a  rectangular  hole  is  cut  through  the 
prism,  the  openings  in  the  front  and  back  faces  will  be  shown  in 
the  development  in  the  centers  of  the  two  broad  faces. 

The  development  of  a  right  prism,  then,  consists  of  as  many 


92 


MECHANICAL     DRAWING. 


rectangles  joined  together  as  the  prism  has  sides,  these  rectangles 
being  the  exact  size  of  the  faces  of  the  prism,  and  in  addition  two 
polygons  the  exact  size  of  the  bases.  It  will  be  found  helpful  in 
developing  a  solid  to  number  or  letter  all  of  the  corners  on  the 

projections,  then 
designate  each  face 
when  developed  in 
the  same  way  as  in 
the  figure. 

If  a  cone  be 
placed  on  its  side  on 
a  plane  surface,  one 
element  will  rest  on 
the  surface.  If  now 
the  cone  be  rolled  on 
the  plane,  the  vertex 
remaining  stationary, 
until  the  same  ele- 
ment is  in  contact 
again,  tha  space  rolled 
over  will  represent 
the  development  of 
the  convex  surface 
of  the  cone.  A,  Fig. 
30,  is  a  cone  cut  by  a 


Fig.  34. 


plane  parallel  to  the 
base.  In  B,  let  the 
vertex  of  the  cone  be 
placed  at  V,  and  one  element  of  the  cone  coincide  with  V  A  I. 
The  length  of  this  element  is  taken  from  the  elevation  A,  of 
either  contour  element.  All  of  the  elements  of  the  cone  are  of 
the  same  length,  so  when  the  cone  is  rolled  each  point  of  the  base 
as  it  touches  the  plane  will  be  at  the  same  distance  from  the 
vertex.  From  this  it  follows  that  the  development  of  the  base 
will  be  the  arc  of  a  circle  of  radius  equal  to  the  length  of  an 
element.  To  find  the  length  of  this  arc  which  is  equal  to  the 
distance  around  the  base,  divide  the  plan  of  the  circumference 
of  the  base  into  any  number  of  equal  parts,  &°  twelve,  then 


MECHANICAL    DRAWING. 


93 


with  the  length  of  one  of  these  parts  as  radius,  lay  off  twelve 
spaces,  1....13,  join  1  and  13  with  V,  and  the  sector  is  the  development 
of  the  cone  from  vertex  to  base.  To  represent  on  the  development 


Fig.  35. 


the  circle  cut  by  the  section  plane,  take  as  radius  the  length  of 
the  element  from  the  vertex  to.  D,  and  with  V  as  center  describe 


Fig.  36. 

an  arc.     The  development  of  the  frustum  of  the  cone  will  be  the 
portion  of  the  circular  ring.     This  of  course  does  not  include  the 


'.'1 


MECHANICAL    DRAWING. 


development  of  the  bases,  which  would  he  simply  two  circles  the 
same  sizes  as  shown  in  plan. 

A  and  B,  Fig.  37,  represent  the  plan  and  elevation  of  a 
regular  triangular  pyramid  and  its  development.  If  face  C  is 
placed  on  the  plane  its  true  size  will  be  shown  at  C  in  the  devel- 
opment. The  true  length  of  the  base  of  triangle  C  is  shown  in 
the  plan.  The  slanting  edges,  however,  not  being  parallel  to  the 
vertical,  are  not  shown  in  elevation  in  their  true  length.  It  be- 
comes necessary  then,  to  find  the  true  length  of  one  of  these  edges 
as  shown  in  Fig.  6,  after  which  the  triangle  may  be  irawn  in  its 
full  size  at  C  in  the  development.  As  the  pyramid  is  regular, 
three  equal  triangles  as  shown  developed  at  C,  D  and  E,  together 
with  the  base  F,  constitute  the  development. 

If  a  right  circular  cylinder  is  to  be  developed,  or  rolled  upon 
a  plane,  the  elements,  being  parallel,  will  appear  as  parallel  lines, 


Pig.  87. 

and  the  base,  being  perpendicular  to  the  elements,  will  develop  as 
a  straight  line  perpendicular  to  the  elements.  The  width  of  the 
development  will  be  the  distance  around  the  cylinder,  or  the  cir- 
cumference of  the  base.  The  base  of  the  cylinder  in  Fig.  38,  is 
divided  into  twelve  equal  parts,  123,  etc.  Commencing  at  point 
1  on  the  development  these  twelve  equal  spaces  are  laid  along 
the  straight  line,  giving  the  development  of  the  base  of  the  cylin- 
der, and  the  total  width.  To  find  the  development  of  the  curve 
cut  by  the  oblique  plane,  draw  in  elevation  the  elements  corre- 
sponding to  the  various  divisions  of  the  base,  and  note  the  points 


MECHANICAL    DRAWING. 


95 


where  they  intersect  the  oblique  plane.  As  we  roll  the  cylinder 
beginning  at  point  1,  the  successive  elements  1,  12,  11,  etc.,  will 
appear  at  equal  distances  apart,  and  equal  in  length  to  the  lengths 
of  the  same  elements  in  elevation.  Thus  point  number  10  on  the 
development  of  the  curve  is  found  by  projecting  horizontally  across 
from  10  in  elevation.  It  will  be  seen  that  the  curve  is  symmetri- 
cal, the  half  on  the  left  of  7  being  similar  to  that  on  the  right. 
The  development  of  any  curve  whatever  on  the  surface  of  the 
cylinder  may  be  found  in  the  same  manner. 

The  principle  of  cylinder  development  is  used  in  laying  out 
elbow  joints,  pipe  ends  cut  off  obliquely,  etc.  In  Fig.  39  is  shown 
plan  and  elevation  of  a  three-piece  elbow  and  collar,  and  develop- 


ments of  the  four  pieces.  In  order  to  construct  the  various  parts 
making  up  the  joint,  it  is  necessary  to  know  what  shape  and  size 
must  be  marked  out  on  the  flat  sheet  metal  so  that  when  cut  out 
and  rolled  up  the  three  pieces  will  form  cylinders  with  the  ends 
fitting  together  as  required.  Knowing  the  kind  of  elbow  desired, 
we  first  draw  the  plan  and  elevation,  and  from  these  make  the 
developments.  Let  the  lengths  of  the  three  pieces  A,  B  and  C 
be  the  same  on  the  upper  outside  contour  of  the  elbow,  the  piece 
B  at  an  angle  of  45°;  the  joint  between  A  and  B  bisects  the 
angle  between  the  two  lengths,  and  in  the  same  way  the  joint 
between  B  and  C.  The  lengths  A  and  C  will  then  be  the  same, 


96 


MECHANICAL    DRAWING. 


and  one  pattern  will  answer  for  both.  The  development  of  A 
is  made  exactly  as  just  explained  for  Fig.  38,  and  this  is  also  the 
development  of  C. 

It  should  be  borne  in  mind  that  in  developing  a  cylinder  we 
must  always  have  a  base  at  right  angles  to  the  elements,  and  if 
the  cylinder  as  given  does  not  have  such  a  base,  it  becomes  neces- 
sary to  cut  the  cylinder  by  a  plane  perpendicular  to  the  elements, 
and  use  the  intersection  as  a  base.  This  point  must  be  clearly 
understood  in  order  to  proceed  intelligently.  A  section  at  right 
angles  to  the  elements  is  the  only  section  which  will  unroll  in  a 


Fig.  30. 

straight  line,  and  is  therefore  the  section  from  which  we  must 
work  in  developing  other  sections.  As  B  has  neither  end  at  right 
angles  to  its  length,  the  plane  X  is  drawn  at  the  middle  and  per- 
pendicular to  the  length.  B  is  the  same  diameter  pipe  as  C  and 
A,  so  the  aection  cut  by  X  will  be  a  circle  of  the  same  diameter 
as  the  base  of  A,  and  its  development  is  shown  at  X. 

From  the  points  where  the  elements  drawn_on   the  elevation 
of  A  meet  the  joint  between  A  and  B,  elements  are  drawn  on  B, 


MECHANICAL    DRAWING. 


'.'7 


which  are  equally  spaced  around  B  the  same  as  on  A.  The  spaces 
then  laid  off  along  X  are  the  same  as  given  on  the  plan  of  A. 
Commencing  with  the  left-hand  element  in  B,  the  length  of  the 
upper  element  between  X  and  the  top  corner  of  the  elbow  is  laid 
off  above  X,  giving  the  first  point  in  the  development  of  the  end 
of  B  fitting  with  C.  The  lengths  of  the  other  elements  in  the 
elevation  of  B  are  measured  in  the  same  way  and  laid  off  from  X. 
The  development  of  the 
other  end  of  the  piece 
B  is  laid  off  below  X, 
using  the  same  distances, 
since  X  is  half  way  be- 
tween the  ends.  The 
development  of  the 
collar  is  simply  the  de. 
velopment  of  the  frus- 
tum of  a  cone,  which  has 
already  been  explained, 
Fig.  36.  The  joint  be- 
tween B  and  C  is  shown 
in  plan  as  an  ellipse,  the 
construction  of  which 
the  student  should  be 
able  to  understand  from 
a  study  of  the  figure. 

The  intersection  of 
a  rectangular  prism  and 

pyramid  is  shown  in  Fig.  40.  The  base  b  c  de  of  the  pyramid  is 
shown  dotted  in  plan,  as  it  is  hidden  by  the  prism.  All  four  edges 
of  the  pyramid  pass  through  the  top  of  the  prism,  1,  2,  3,  4.  As 
the  top  of  the  prism  is  a  horizontal  plane,  the  edges  of  the  pyramid 
are  shown  passing  through  the  top  in  elevation  at  x*  g»  fa  i».  These 
four  points  might  be  projected  to  the  plan  on  the  four  edges  of  the 
pyramid;  but  it  is  unnecessary  to  project  more  than  one,  since  the 
general  principle  applies  here  that  if  a  cone,  pyramid,  prism  or 
Cylinder  be  cut  by  a  plane  parallel  to  the  base,  the  section  is  a 
figure  parallel  and  similar  to  the  base.  The  one  point  x*  is  there- 
fore projected  down  to  a  b  in  plan,  giving  x\  and  with  this  a«j 


Fig.  40. 


MECHANICAL    DRAWING. 


one  corner,  the  square  xh  gjl  ih  kh  is  drawn,  its  sides  parallel  to'  the 
edges  of  the  base.  This  square  is  the  intersection  of  the  pyramid 
with  the  top  of  the  prism. 

The  intersection  of  the  pyramid  with  the  bottom  of  the  prism 
is  found  in  like  manner,  by  taking  the  point  where  one  edge  of 
the  pyramid  as  a  b  passes  through  the  bottom  of  the  prism  shown 
in  elevation  as  point  W,  projecting  down  to  w>*  on  ah  £/*,  and 
drawing  the  square  mh  nh  oh  ph  parallel  to  the  base  of  the  pyramid. 
These  two  squares  constitute  the  entire  intersection  of  the  two 
solids,  the  pyramid  going  through  the  bottom  and  coming  out  at 
the  top  of  the  prism.  As  much  of  the  slanting  edges  of  the 


10  7 


Fig.  41. 

pyramid  as  are  above  the  prism  will  be  seen  in  plan,  appearing  as 
the  diagonals  of  the  small  square,  and  the  rest  of  the  pyramid, 
being  below  the  top  surface  of  the  prism,  will  be  dotted  in  plan. 
Fig.  41  is  the  development  of  the  rectangular  prism,  show- 
ing the  openings  in  the  top  and  bottom  surfaces  through  which 
the  pyramid  passed.  The  development  of  the  top  and  bottom, 
back  and  front  faces  will  be  four  rectangles  joined  together,  the 
same  sizes  as  the  respective  faces.  Commencing  with  the  bottom 
face  5678,  next  would  come  the  back  face  6127,  then  the  top, 
etc.  The  rectangles  at  the  ends  of  the  top  face  1  2  3  4  are  the 
ends  of  the  prism.  These  might  have  been  joined  on  any  other 


MECHANICAL    DRAWING.  99 


face  as  well.  Now  find  the  development  of  the  square  in  the  bottom 
5678.  As  the  size  will  be  the  same  as  in  projection,  it  only  re- 
mains to  determine  its  position.  This  position,  however,  will 
have  the  same  relation  to  the  sides  of  the  rectangle  as  in  the  plan. 
The  center  of  the  square  in  this  case  is  in  the  center  of  the  face. 
To  transfer  the  diagonals  of  the  square  to  the  development,  extend 
them  in  plan  to  intersect  the  edges  of  the  prism  in  points  9,  10, 
11  and  12.  Take  the  distance  from  5  to  9  along  the  edge  5  6, 
and  lay  it  on  the  development  from  5  along  5  6,  giving  point  9. 
Point  10  located  in  the  same  way  and  connected  with  9,  gives  the 
position  of  one  diagonal.  The  other  diagonal  is  obtained  in  a 
similar  way,  then  the  square  constructed  on  these  diagonals.  The 
same  method  is  used  for  locating  the  small  square  on  the  top  face. 

If  the  intersection  of  a  cylinder  and  prism  is  to  be  found,  we 
may  either  obtain  the  points  where  elements  of  the  cylinder  pierce 
the  prism,  or  where  edges  and  lines  parallel  to  edges  on  the  sur- 
face of  the  prism  cut  the  cylinder. 

A  series  of  parallel  planes  may  also  be  taken  cutting  curves 
from  the  cylinder  and  straight  lines  from  the  prism ;  the  intersec- 
tions give  points  on  the  intersection  of  the  two  solids. 

Fig.  42  represents  a  triangular  prism  intersecting  a  cylinder. 
The  axis  of  the  prism  is  parallel  to  V  and  inclined  to  H.  Starting 
with  the  size  and  shape  of  the  base,  this  is  laid  off  at  a,  bh  ch,  and 
the  altitude  of  the  triangle  taken  and  laid  off  at  av  cv  in  elevation, 
making  right  angles  with  the  inclination  of  the  axis  to  H.  The 
plan  of  the  prism  is  then  constructed.  To  find  the  intersection  of 
the  two  solids,  lines  are  drawn  on  the  surface  of  the  prism  parallel 
to  the  length  and  the  points  where  these  lines  and  the  edges 
pierce  the  cylinder  are  obtained  and  joined,  giving  the  curve. 

The  top  edge  of  the  prism  goes  into  the  top  of  the  cylinder. 
This  point  will  be  shown  in  elevation,  smce  the  top  of  the  cylinder 
is  a  plane  parallel  to  H  and  perpendicular  to  V,  and  therefore 
projected  on  V  as  a  straight  line.  The  upper  edge,  then,  is  found 
to  pass  into  the  top  of  the  cylinder  at  point  o,  ov  and  oh.  The 
intersection  of  the  two  upper  faces  of  the  prism  with  the  top  of 
the  cylinder  will  be  straight  lines  drawn  from  point  o  and  will  be 
shown  in  plan.  If  we  can  find  where  another  line  of  the  surface 
o  a  h  14  pierces  the  upper  base  of  the  cylinder,  this  point  joined 


100 


MECHANICAL     DRAWING. 


ivith  o  will  determine  the  intersection  of  this  face  with  the  top  of 
the  cylinder.  A  surface  may  always  be  produced,  if  necessary, 
to  find  an  intersection. 

Edge  a  b  pierces  the  plane  of  the  top  of  the  cylinder  at  point 


d,  seen  in  elevation  ;  therefore  the  line  joining  this  point  with  o  is 
the  intersection   of  one  upper  face  of  the  prism  with   the  upper 


MECHANICAL     DRAWING.  101 

base  of  the  cylinder.  The  only  part  of  this  line  needed,  of  course, 
is  within  the  actual  limits  of  the  base,  that  is  o  9.  The  intersec- 
tion o  8  of  the  other  top  face  is  found  by  tine  same  method.  On 
the  convex  surface  of  the  cylinder  there  will  be  three  curves,  one 
for  each  face  of  the  prism.  Points  b  and  9  on  the  upper  base  of 
the  cylinder,  will  be  where  the  curves  for  the  two  upper  faces  will 
begin.  The  point  d  is  found  on  the  revolved  position  of  the  base 
at  dr  and  d{  b  is  divided  into  the  equal  parts  d{  —  er  e{  — /j,  etc., 
which  revolve  back  to  dh,  eh,fh  and  <jh.  The  divisions  are  made 
equal  merely  for  convenience  in  developing.  The  vertical  pro- 
jections of  f?,  e,  etc.,  are  found  on  the  vertical  projection  of  «  />, 
directly  above  d\  eh,  etc.,  or  may  be  found  by  taking  from  the 
revolved  position  of  the  base,  the  perpendiculars  from  <7,  <?,  etc.,  to 
<:h  bh  and  laying  them  off  in  elevation  from  lv  along  bv  <iv.  Lines 
such  as/ 12,  m  5,  etc.,  parallel  to  a  o  are  drawn  in  plan  and  eleva- 
tion. Points  ih  kh  mh  nh  are  taken  directly  behind  dh  >ih  fh  yh 
hence  their  vertical  projections  coincide.  Points  •;?,  w,  k\  and  z(  are 
formed  by  projecting  across  from  nh  mh  kh  and  I'1. 

The  convex  surface  of  the  cylinder  is  perpendicular  to  H,  so 
the  points  where  the  lines  on  the.prism  pierce  it  will  be  projected 
on  plan  as  the  points  where  these  lines  cross  the  circle,  14,  13,12, 

11 3.      The  vertical  projections  of  these  points  are  found  on 

the  corresponding  lines  in  elevation,  and  the  curves  drawn  through. 
The  curve  3,  4.. ..8  must  be  dotted,  as  it  is  on  the  back  of  the 
cylinder.  The  under  face  of  the  prism,  which  ends  with  the  line 
b  c,  is  perpendicular  to  the  vertical  plane,  so  the  curve  of  intersec 
tion  will  be  projected  on  V  as  a  straight  line.  Point  14  is  one 
end  of  this  curve.  3  the  other  end,  and  the  curve  is  projected  in 
elevation  as  the  straight  line  from  14  to  the  point  where  the  lower 
edge  of  the  prism  crosses  the  contour  element  of  the  cylinder. 

Fig.  43  gives  the  development  of  the  right-hand  half  of  the 
cylinder,  beginning  with  number  1.  As  previously  explained,  the 
distance  between  the  elements  is  shown  in  the  plan,  as  1 — 2,  2 — 3, 

3 4  and  so  on.     These   spaces  are  laid  off  in  the  development 

along  a  straight  line  representing  the  development  of  the  base, 
and  from  these  points  the  elements  are  drawn  perpendicularly. 

The  lengths  of  the  elements  in  the  development  from  the  base 
to  the  curve  are  exactly  the  same  as  on  the  elevation,  as  the 


102 


MECHANICAL     DRAWING. 


elevation  gives  the  true  lengths.  If  then  the  development  of  the 
base  is  laid  off  along  the  same  straight  line  as  the  vertical  projec- 
tion of  the  base,  the  points  in  elevation  may  be  projected  across 
with  the  T-square  to  the  corresponding  elements  in  the  develop- 
ment. The  points  on  the  curve  cut  by  the  under  face  of  the 
prism  are  on  the  same  elements  as  the  other  curves,  and  their 
vertical  projections  are  on  the  under  edge  of  the  prism,  hence 
these  points  are  projected  across  for  the  development  of  the  lower 
curve. 

In  Fig.  44  is  given  the  development  of  the  prism  from  the 
right-hand  end  as  far  as  the  intersection  with  the  cylinder,  begin- 

14 


Fig.  44. 

ning  at  the  left  with  the  top  edge  a  0,  the  straight  line  a  b  e  a 
being  the  development  of  the  base.  As  this  must  be  the  actual 
distance  around  the  base,  the  length  is  taken  from  the  true  size 
of  the  base,  a,  bh  ch.  The  parallel  lines  drawn  on  the  surfaces  of 
the  prism  must  appear  on  the  development  their  true  distances 
apart,  hence  the  distances  a,  <?,,  rf,  et,  etc.,  are  made  equal  to 
ad,de,  etc.  on  the  development.  The  actual  distances  between  the 
parallel  lines  on  the  bottom  face  of  the  prism  are  shown  along 
the  edge  of  the  base,  bh  ch.  Perpendicular  lines  are  drawn  from 
the  points  of  division  on  the  development. 

The  position  of  the  developed  curve  is  found  by  laying  off 
the  true  lengths  on  the  perpendiculars.  These  true  lengths  (of 
the  parallel  lines)  are  not  shown  in  plan,  as  the  lines  are  not 
parallel  to  the  horizontal  plane,  but  are  found  in  elevation.  The 
length  oa  on  the  development,  is  equal  to  av  ovt  d  10  to  dv  10,  and 


MECHANICAL   DRAWING 


103 


so  on  for  all  the  rest.  Point  9  is  found  as  follows:  on  the  projec- 
tions, the  straight  line  from  o  to  d  passes  through  point  9  and  the 
true  distance  from  o  to  9  is  shown  in  plan.  All  that  is  necessary, 
then,  is  to  connect  o  and  d  on  the  development,  and  lay  off  from  o 
the  distance  o"9.  Number  8  is  found  in  the  same  way. 

ISOMETRIC  PROJECTION. 

Heretofore  an  object  has  been  represented  by  two  or  more 
projections.  Another  system,  called  isometrical  drawing,  is  used 
to  show  in  one  view  the  three  dimensions  of  an  object,  length  (or 
height),  breadth,  and  thickness.  An  isometrical  drawing  of  an 
object,  as  a  cube,  is  called  for  brevity  the  "  isometric  "  of  the  cube. 


Fig.  45. 

To  obtain  a  view  which  shows  the  three  dimensions  in  such  a 
way  that  measurements  can  be  taken  from  them,  draw  the  cube  in 
the  simple  position  shown  at  the  left  of  Fig.  45,  in  which 
it  rests  on  H  with  two  faces  parallel  to  V;  the  diagonal  from  the 
front  upper  right-hand  corner  to  the  back  lower  left-hand  corner  is 
indicated  by  the  dotted  line.  Swing  the  cube  around  until  the 
diagonal  is  parallel  with  V  as  shown  in  the  second  position.  Here 
the  front  face  is  at  the  right.  In  the  third  position  the  lower  end 
of  the  diagonal  has  been  raised  so  that  it  is  parallel  to  H,  becoming 
thus  parallel  to  both  planes.  The  plan  is  found  by  the  principles 
of  projection,  from  the  elevation  and  the  preceding  plan.  The  front 
face  is  now  the  lower  of  the  two  faces  shown  in  the  elevation. 
From  this  position  the  cube  is  swung  around,  using  the  corner 


104 


MECHANICAL   DRAWING 


resting  on  the  H  as  a  pivot,  until  the  diagonal  is  perpendiculai 
to  V  but  still  parallel  to  H.  The  plan  remains  the  same,  except  as 
regards  position;  while  the  elevation,  obtained  by  projecting  across 
from  the  previous  elevation,  gives  the  isometrical  projection  of  the 
cube.  The  front  face  is  now  at  the  left. 

In  the  last  position,  as  one  diagonal  is  perpendicular  to  V,  it 
follows  that  all  the  faces  of  the  cube  make  equal  angles  with  V, 
hence  are  projected  on  that  plane  as  equal  parallelograms.  For  the 
same  reason  all  the  edges  of  the  cube  are  projected  in  elevation  in 
equal  lengths,  but,  being  inclined  to  V,  appear  shorter  than  they 
actually  are  on  the  object.  Since  they  are  all  equally  foreshortened 
and  since  a  drawing  may  be  made  at  any  scale,  it  is  customary  to 

make  all  the  isometrical  lines  of  a 
drawing  full  length .  Th  is  will  give 
the  same  proportions,  and  is  much 
the  simplest  method.  Herein  lies 
the  distinction  between  an  isomet- 
rical projection  and  an  isometric 
drawing. 

It  will  be  noticed  that  the 
figure  can  be  inscribed  in  a  circle, 
and  that  the  outline  is  a  perfect 
hexagon.  Hence  the  lines  showing 
breadth  and  length  are  30°  lines, 
while  those  showing  height  are 
vertical. 

Fig.  46  shows  the  isometric  of  a  cube,  1  inch  square.  All  of 
the  edges  are  shown  in  their  true  length,  hence  all  the  surfaces 
appear  of  the  same  size.  In  the  figure  the  edges  of  the  base  are 
inclined  at  30°  with  a  T-square  line,  but  this  is  not  always  the  case. 
For  rectangular  objects,  such  as  prisms,  cubes,  etc.,  the  base 
edges  are  at  30°  only  when  the  prism  or  cube  is  supposed  to  be  in 
the  simplest  possible  position.  The  cube  in  Fig.  46  is  supposed  to 
be  in  the  position  indicated  by  plan  and  elevation  in  Fig.  47,  that 
is,  standing  on  its  base,  with  two  faces  parallel  to  the  vertical 
plane. 

If  the  isometric  of  the  cube  in  the  position  of  Fig.  48  were 
required,  it  could  not  be  drawn  with  the  base  edges  at  30°;  neither 


Fig.  46. 


MECHANICAL   DKAWING 


105 


would  these  edges  appear  in  their  true  lengths.  It  follows,  then, 
that  in  isometrical  drawing,  true  lengths  appear  only  as  30°  lines 
or  as  vertical  lines.  Edges  or  lines  that  in  actual  projection  are 
either  parallel  to  the  ground  line  or  perpendicular  to  V,  are  drawn 
in  isometric  as  30°  lines,  full  length;  and  those  that  are  actually 
vertical  are  made  vertical  in  isometric,  also  full  length. 

In  Fig.  45,  lines  such  as  the  front  vertical  edges  of  the  cube 
and  the  two  base  edges  are  called  the  three  isometric  axes.  The 
isometric  of  objects  in  oblique  positions,  as  in  Fig.  48,  can  be  con- 


Pig.  47.  Pig.  48. 

structed  only  by  reference  to  their  projections,  by  methods  which 
will  be  explained  later. 

In  isometric  drawing  small  rectangular  objects  are  more  satis- 
factorily represented  than  large  curved  ones.  In  woodwork,  mor- 
tises and  joints  and  various  parts  of  framing  are  well  shown  in 
isometric.  This  system  is  used  also  to  give  a  kind  of  bird's-eye 
view  of  the  mills  or  factories.  It  is  also  used  in  making  sketches 
of  small  rectangular  pieces  of  machinery,  where  it  is  desirable  to 
give  shape  and  dimensions  in  one  view. 

In  isometric  drawing  the  direction  of  the  ray  of  light  is 
parallel  to  that  diagonal  of  a  cube  which  runs  from  the  upper  left 
corner  to  the  lower  right  corner,  as  4V-7V  in  the  last  elevation  of 
Fig.  45.  This  diagonal  is  at  30° ;  hence  in  isometrical  drawing 
the  direction  of  the  light  is  at  30°  downward  to  the  right.  From 


331 


100 


MECHANICAL   DRAWING 


this  it  follows  that  tho  top  and  two  left-hand  faces  of  the  cube  are 
light,  the  others  dark.     This  explains  the  shade  lines  in  Fig. '45. 

In  Fig.  45,  the  top  end  of  the  diagonal  which  is  parallel  to  the 
ray  of  light  in  the  first  position  is  marked  4,  and  traced  through 
to  the  last  or  isometrical  projection,  4V.  It  will  be  seen  that  face 
3V  4V  5V  8V  of  the  isometric  projection  is  the  front  face  of  the  cube 
in  the  first  view;  hence  we  may  consider  the  left  front  face  of  the 
isometric  cube  as  the  front.  This  is  not  absolutely  necessary, 
but  by  so  doing  the  isometric  shade  edges  are  exactly  the  same 
as  on  the  original  projection. 


A  It- 


Fig.  49.  Fig.  50. 

Fig.  49  shows  a  cube  with  circles  inscribed  in  the  top  and 
two  side  faces.  The  isometric  of  a  circle  is  an  ellipse,  the  exact 
construction  of  which  would  necessitate  finding  a  number  of  points; 
for  this  reason  an  approximate  construction  by  arcs  of  circles  is 
often  made.  In  the  method  of  Fig.  49,  four  centers  are  used. 
Considering  the  upper  face  of  the  cube,  lines  are  drawn  from  the 
obtuse  angles/* and  e^  to  the  centers  of  the  opposite  sides. 

The  intersections  of  these  lines  give  points  g  and  ^,  which 
serve  as  centers  for  the  ends  of  the  ellipse.  With  center  g  and 
radius  g  a,  the  arc  a  d  is  drawp;  and  with/  as  center  and  radius 
fd,  the  arc  d  c  is  described,  and  the  ellipse  finished  by  using 
centers  h  and  e.  This  construction  is  applied  to  all  three  faces. 

Fig.  50  is  the  isometric  of  a  cylinder  standing  on  its  base. 


332 


MECHANICAL   DRAWING 


107 


Notice  that  the  shade  line  on  the  top  begins  and  ends  where 

T-square  lines  would  be  tangent  to  the  curve,  and  similarly  in  the 

case  of  the  part  shown  on  the  base.     The  explanation  of  the  shade 

is  very  similar  to  that  in  pro-  <*, 

jections.  Given  in  projections 

a   cylinder   standing    on    its 

base,  the  plan  is  a  circle,  and 

the  shade  line  is  determined 

by  applying  the  45°  triangle 

tangent  to  the  circle.     This  is 

done  because  the  45°  line  is 

the  projection  of  the  ray  of 

light    on  the   plane   of     the 

base. 

In  Fig.  49,  the  diagonal  m  I  may  represent  the  ray  of  light 
and  its  projection  on  the  base  is  seen  to  be  k  /,  the  diagonal  of  the 
base,  a  T-square  line.  Hence,  for  the  cylinder  of  Fig.  50,  apply 
tangent  to  the  base  and  also  to  the  top  a  line  parallel  to  the 
projection  of  the  ray  of  light  on  these  planes,  that  is,  a  T-square 
line,  and  this  will  mark  the  beginning  and  ending  of  the  shade  line. 
In  Fig.  49  the  projection  of  the  ray  of  light  diagonal  m  I  on 
the  right-hand  face  is  e  Z,  a  30° 
line;  hence,  in  Fig.  51,  where  the 
base  is  similarly  placed,  apply 
the  30°  triangle  tangent  as  indi- 
cated, determining  the  shade  line 
of  the  base.  If  the  ellipse  on 
the  left-hand  face  of  the  cube  were 
the  base  of  a  cone  or  cylinder 
extending  backward  to  the  right, 
the  same  principle  would  be  used. 

The  projection  of  the  cube  diagonal  m  I  on  that  face  is  m  n,  a 
60°  line;  hence  the  60°  triangle  would  be  used  tangent  to  the  base 
in  this  last  supposed  case,  giving  the  ends  of  the  shade  line  at 
points  o  and  r.  Figs.  52,  53  and  54  illustrate  the  same  idea  with 
respect  to  prisms,  the  direction  of  the  projection  of  the  ray  of  light 
on  the  plane  of  the  base  being  used  in  each  case  to  determine  the 
light  and  dark  faces  and  hence  the  shade  lines. 


Fig.  52. 


333 


108 


MECHANICAL   DRAWING 


In  Fig.  52  a  prism  is  represented  standing  on  its  base,  so  that 

the  projection  of  the  cube  diagonal  on  the  base  (that  is,  a  T-square 

line)   is  used  to  determine  the  light  and  dark  faces  as  shown. 

The  prism  in  Fig.  53  has  for 
its  base  a  trapezium.  The 
projection  of  the  ray  of  light 
on  this  end  is  parallel  to  the 
diagonal  of  the  face;  hence 
the  60°  triangle  applied  par- 
allel to  this  diagonal  shows 
that  faces  a  c  d  I  and  a  g  hi 
are  light,  while  c  ef  d  and 
g  e  f  h  are  dark,  hence  the 
shade  lines  as  shown. 

The  application  in  Fig. 
54    is    the    same,    the    only 

difference  being  in  the  position  of  the  prism,  and  the  consequent 

difference  in  the  direction  of  the  diagonal. 

Fig.  55  represents  a  block  with  smaller  blocks  projecting  from 

three  faces. 

Fig.  56  shows  a  framework  of  three  pieces,  two  at  right  angles 

and  a  slanting  brace.     The  horizontal  piece  is  mortised  into  the 

upright,  as  indicated  by  the 

dotted  lines.     In  Fig.  57 

the  isometric  outline  of  a 

house  is  represented,  show- 
ing a  dormer  window  and 

a  partial  hip  roof;  a  I  is  a 

hip  rafter,  c  d  a  valley.   Let 

the  pitch  of  the  main  roof 

be  shown  at  B,  and  let  m  be 

the  middle  point  of  the  top 

of    the    end  wall  of    the 

house.  Then,  by  measuring 

vertically  up  a  distance  m  I 

equal  to  the  vertical  height 


Fig.  54. 


a  n  shown  at  B,  a  point  on  the  line  of  the  ridge  will  be  found  at  I. 
Line  I  i  is  equal  to  I  h,  and  i  li  is  then  drawn.     Let  the  pitch  of 


334 


MECHANICAL   DRAWING 


109 


the  end  roof  be  given  at  A.  This  shows  that  the  peak  of  the  roof, 
or  the  end  a  of  the  ridge,  will  be  back  from  the  end  wall  a  distance 
equal  to  the  base  of  the  triangle  at  A.  Hence  lay  off  from  I  this 
distance,  giving  point  #,  and  join  a  with  5  and  x. 


Fig.  57. 


The  height  k  e  of  the  ridge  of  the  dormer  roof  is  known,  and 
we  must  find  where  this  ridge  will  meet  the  main  roof.  The  ridge 
must  be  a  30°  line  as  it  runs  parallel  to  the  end  wall  of  the  hou 


no 


MECHANICAL   DRAWING 


and  to  the  ground.  Draw  from  e  tt  line  parallel  to  J  h  to  meet  a 
vertical  through  h  atf.  This  point  is  in  the  vertical  plane  of  the 
end  wall  of  the  house,  hence  in  the  plane  of  i  h.  If  now  a  30°  line 
be  drawn  from/"  parallel  to  x  5,  it  will  meet  the  roof  of  the  house 
at  g.  The  dormer  ridge  andfg  are  in  the  same  horizontal  plane, 
hence  will  meet  the  roof  at  the  same  distance  below  the  ridge  a  i. 
Therefore  draw  the  30°  line  g  c,  and  connect  c  with  d. 

In  Fig.  58  a  box  is  shown  with  the  cover  opened  through  150°. 


Fig.  58. 

The  right-hand  edge  of  the  bottom  shows  the  width,  the  left-hand 
edge  the  length,  and  the  vertical  edge  the  height.  The  short  edges 
of  the  cover  are  not  isometric  lines,  hence  are  not  shown  in  their 
true  lengths;  neither  is  the  angle  through  which  the  cover  is  opened 
represented  in  its  actual  size. 

The  corners  of  the  cover  must  then  be  determined  by  co- 
ordinates from  an  end  view  of  the  box  and  cover.  As  the  end  of 
the  coyer  is  in  the  same  plane  as  the  end  of  the  box,  the  simple 


336 


MECHANICAL  DRAWING 


111 


end  view  as  shown  in  Fig.  59  will  be  sufficient.  Extend  the  top  of 
the  box  to  the  right,  and  from  c  and  d  let  fall  perpendiculars  or 
a  b  produced,  giving  the  points  e  and  /.  The  point  c  may  be 
located  by  means  of  the  two  distances  or  co-ordinates  I  e  and  e  c. 


Fig.  59. 

and  these  distances  will  appear  in  their  true  lengths  in  the 
isometric  view.  Hence  produce  a'  I'  to  e'  and/' ;  and  from  these 
points  draw  verticals  e'  c'  and/*  d1 ';  make  I'  e'  equal  to  5  e,  e'  c' 
equal  to  e  c;  and  similarly  for  d' '.  Draw  the  lower  edge  parallel 
to  c'  d'  and  equal  to  it  in  length,  and  r-, 
connect  with  5'. 

It  will  be  seen  that  in  isometric  draw- 
ing parallel  lines  always  appear  parallel. 
It  is  also  true  that  lines  divided  propor- 
tionally maintain  this  same  relation  in 
isometric  drawing. 

Fig.  60  shows  a  block  or  prism  with  a 
semicircular  top.  Find  the  isometric  of 
the  square  circumscribing  the  circle,  then 
draw  the  curve  by  the  approximate  method. 
The  centers  for  the  back  face  are  found 
by  projecting  the'  front  centers  back  30° 
equal  to  the  thickness  of  the  prism,  as 
shown  at  a  and  b.  The  plan  and  elevation  of  an  oblique  pentagonal 
pyramid  are  shown  in  Fig.  61.  It  is  evident  that  none  of  the 
edges  of  the  pyramid  can  be  drawn  in  isometric  as  either  vertical 
or  30°  lines;  hence,  a  system  of  co-ordinates  must  be  used  as 


Fig.  60. 


337 


112 


MECHANICAL  DRAWING 


shown  in  Fig.  58.  This  problem  illustrates  the  most  general  case; 
and  to  locate  some  of  the  points  three  co-ordinates  must  be  used, 
two  at  30°  and  one  vertical. 

Circumscribe,  about  the  plan  of  the  pyramid,  a  rectangle  which 
shall  have  its  sides  respectively  parallel  and  perpendicular  to  the 
ground  line.  This  rectangle  is  on  H,  and  its  vertical  projection  is 
in  the  ground  line. 

The  isometric  of  this  rectangle  can  be  drawn  at  once  with  30° 
lines,  as  shown  in  Fig.  62,  o  being  the  same  point  in  both  figures. 


Fig.  62. 


Fig.  61. 

The  horizontal  projection  of  point  3  is  found  in  isometric  at  3h,  at 
tho  same  distance  from  o  as  in  the  plan.  That  is,  any  distance 
which  in  plan  is  parallel  to  a  side  of  the  circumscribing  rectangle, 
is  shown  in  isometric  in  its  true  length  and  parallel  to  the  corre- 
sponding side  of  the  isometric  rectangle.  If  point  3  were  on  the 
horizontal  plane  its  isometric  would  be  3h,  but  the  point  is  at  the 
vertical  height  above  H  given  in  the  elevation;  hence,  lay  off  above 
3h  this  vertical  height,  obtaining  the  actual  isometric  of  the  point. 
To  locate  4,  draw  4  a  parallel  to  the  side  of  the  rectangle;  then  lay 


338 


MECHANICAL   DRAWING 


113 


Fig.  63. 


off  o  a,  and  a  4h,  giving  what  may  be  called  the  isometric  plan  of  4 
Next,  the  vertical  height  taken  from  the  elevation  locates  the  iso- 
metric of  the  point  in  space. 
In    like    manner  all  the 
corners  of  the   pyramid,  in, 
eluding  the  apex,  are  located. 
The   rule   is,  locate  -first  in 
isometric  the  horizontal  pro- 
jection of  a  point  fiy  one  or 
two    30°   co-ordinates;    then 
vertically,  above    this  point, 
its    height    as    taken   from 
the    elevation.      The    shade 
lines   cannot   be  determined 
here  by  applying  the  30°  or 
(50°    triangle,   owing    to    the 
obliquity  of  the  faces.     Since 
the  right  front  corner  of  the 
rectangle  in  plan  was  made  the  point  o  in  isometric,  the  shade 
lines  must  be  the  same  in  isometric  as  in  actual  projection;  so  that, 

if  these  can  be  de- 
termined in  Fig.  61, 
they  may  be  applied 
at  once  to  Fig.  62. 
The  shade  lines 
in  Fig.  61  are  found 
by  a  short  method 
which  is  convenient 
to  use  when  the  exact 
shade  lines  are  de- 
sired, and  when  they 
cannot  be  deter- 
mined by  applying 
the  45°  triangle.  A 
plane  is  taken  at  45° 

Fig.  64.  with  the  horizontal 

plane,  and  parallel  to  the  direction  of  the  ray  of  light,  in  such  a 
position  as  to  cut  all  the  surfaces  of  the  pyramid,  as  sho»n  m 


114 


MECHANICAL   DRAWING 


elevation.  This  plane  is  perpendicular  to  the  vertical  plane;  hence 
the  section  it  cuts  from  the  pyramid  is  readily  found  in  plan  by 
projection.  This  plane  contains  some  of  the  rays  of  light  falling 
upon  the  pyramid;  and  we  can  tell  what  surfaces  these  rays  strike 


Q 


Fig.  65. 


Fig.  67. 


Fig.  68. 

and  make  light,  by  noticing  on  the  plan  what  edges  of  the  section  are 
struck  by  the  projections  of  the  rays  of  light.  That  is,  r  s,  s  t,  and  t  u 
receive  the  rays  of  light;  hence  the  surfaces  on  which  these  lines  lie 
are  light.  T  s  is  on  the  surface  determined  by  the  two  lines  passing 


340 


MECHANICAL   DRAWING 


115 


through  r  and  «,  namely,  2  —  1  and  1  —  5;  in  other  words,  r  s  is 
on  the  base;  similarly,  s  t  is  on  the  surface  1  —  5  —  6;  and  t  u  on 
the  surface  4  —  6  —  5.  The  other  surfaces  are  dark;  hence  the  edges 
which  are  between  the  light  and  dark  faces  are  the  shade  lines. 

Whenever  it  is  more  convenient,  a  plane  parallel  to  the  ray 
of  light  and  perpendicular  to  H  may  be  taken,  the  section  found 
in  elevation,  and  the  45°  triangle  applied  to  this  section.  The 
same  method  may  be  used  to  determine  the  exact  shade  lines 
of  a  cone  or  cylinder  in  an  oblique  position. 

Figs.  63  to  70  give  examples  of  the  isometric  of  various 
objects.  Fig.  65  is  the  plan  and  elevation,  and  Fig.  66  the 


Fig.  69.  FiS-  70- 

isometric,  of  a  carpenter's  bench.  In  Fig.  70,  take  especial  notice 
of  the  shade  lines.  These  are  put  on  as  if  the  group  were  made 
in  one  piece;  and  the  shadows  cast  by  the  blocks  on  one  another 
are  disregarded.  All  upper  horizontal  faces  are  light,  all  left-hand 
(front  and  back)  faces  light,  and  the  rest  dark. 

OBLIQUE    PROJECTIONS. 

In  oblique  projection,  as  in  isometric,  the  end  sought  for  is 
the  same-a  more  or  Isss  complete  representation,  in  one  view,  of 
any  object.  Oblique  projection  differs  from  isometric  in  that  one 
face  of  the  object  is  represented  as  if  parallel  to  the  vertical 
plane  of  projection,  the  others  inclined  to  it.  Another  pon 


341 


110 


MECHANICAL   DRAWING 


difference   is   that  oblique    projection    cannot   be  deduced  from 
orthographic  projection,  as  is  isometric. 

In  oblique  projection  all  lines  in  the  front  face  are  shown  in 
their  true  lengths  and  in  their  true  relation  to  one  another,  and 
lines  which  are  perpendicular  to  this  front  face  are  shown  in  their 
true  lengths  at  any  angle  that  may  be  desired  for  any  particular 
case.  Lines  not  in  the  plane  of  the  front  face  nor  perpendicular 


Fig.  71. 


Fig.  72. 


to  it  must  be  determined  by  co-ordinates,  as  in  isometric.  It  will 
be  seen  at  once  that  this  system  possesses  some  advantages  over 
the  isometric,  as,  for  instance,  in  the  representation  of  circles, 


45 


Fig.  73. 


Fig.  74. 


as  any  circle  or  curve  in  the  front  face  is  actually  drawn  as  such. 
The  rays  of  light  are  still  supposed  to  be  parallel  to  the  same 
diagonal  of  the  cube,  that  is,  sloping  downward,  toward  the  plane 
of  projection,  and  to  the  right,  or  downward,  backward  and  to 
the  right.  Figs_71,  72  and  73  show  a  cube  in  oblique  projection, 


342 


MECHANICAL   DRAWING 


117 


with  the  30°,  45°  and  60°  slant  respectively.  The  dotted  diagonal 
represents  for  each  case  the  direction  of  the  light,  and  the  shade 
lines  follow  from  this. 

The  shade  lines  have  the  same  general  position  as  in  isometric 


Fig.  75.  Fig.  76. 

drawing,  the  top,  front  and  left-hand  faces  being  light.  No  matter 
what  angle  may  be  used  for  the  edges  that  are  perpendicular  to 
the  front  face,  the  projection  of  the  diagonal  of  the  cube  on  this 
face  is  always  a  45°  line;  hence,  for  determining  the  shade  lines  on 


Fig.  77. 

any  front  face,  such  as  the  end  of  the  hollow  cylinder  in  Fig.  74, 
the  45°  line  is  used  exactly  as  in  the  elevation  of  ordinary 
projections. 

Figs.  75,  76,  77  and  79  are  other  examples  of  oblique  projections. 
Fig.  77  is  a  crank  arm. 

The  method  of  using  co-ordinates  for  lines  of  which  the  true 


118 


MECHANICAL   DRAWING 


lengths  are  not  shown,  is  illustrated  by  Figs.  78  and  79.  Fig.  79 
represents  the  oblique  projection  of  the  two  joists  shown  in  plan 
and  elevation  in  Fig.  78".  The  dotted  lines  in  the  elevation  (see 
Fig.  78)  show  the  heights  of  the  corners  above  the  horizontal 
stick.  The  feet  of  these  perpendiculars  give  the  horizontal  dis- 
tances of  the  top  corners  from  the  end  of  the  horizontal  piece. 

In  Fig.  79  lay  off  from  the  upper  right-hand  corner  of  the 
front  end  a  distance  equal  to  the  distance  between  the  front  edge 
of  the  inclined  piece  and  the  front  edge  of  the  bottom  piece  (see 
Fig.  78).  From  this  point  draw  a  dotted  line  parallel  to  the 


Fig.  79 


Fig.  78. 

length.  The  horizontal  distances  from  the  iipper  left  corner  to 
the  dotted  perpendicular  are  then  marked  off  on  this  line.  From 
these  points  verticals  are  drawn,  and  made  equal  in  length  to  the 
dotted  perpendiculars  of  Fig.  78,  thus  locating  two  corners  of  the 
end. 

LINE  SHADING. 

In  finely  finished  drawings  it  is  frequently  desirable  to  make 
the  various  parts  more  readily  seen  by  showing  the  graduations  of 
light  and  shade  on  the  curved  surfaces.  This  is  especially  true  of 
such  surfaces  as  cylinders,  cones  and  spheres.  The  effect  is 
obtained  by  drawing  a  series  of  parallel  or  converging  lines  on 
the  surface  at  varying  distances  from  one  another.  Sometimes 
draftsmen  vary  the  width  of  the  lines  themselves.  These  lines  are 
farther  apart  on  the  lighter  portion  of  the  surface,  and  are  closer 
together  and  heavier  on  the  darker  part. 


344 


MECHANICAL   DRAWING 


Fig.  80  shows  a  cylinder  with  elements  drawn  on  the  surface 
equally  spaced,  -as  on  the  plan.  On  account  of  the  curvature  of 
the  surface  the  elements  are  not  equally  spaced  on  the  elevation, 
but  give  the  effect  of  graduation  of  light.  The 
result  is  that  in  elevation  the  distances  between 
the  elements  gradually  lessen  from  the  center 
toward  each  side,  thus  showing  that  the  cylinder 
is  convex.  The  effect  is  intensified,  however,  if 
the  elements  are  made  heavier,  as  well  as  closer 
together,  as  shown  in  Figs.  81  to  87. 

Cylinders  are  often  shaded  with  the  light 
coming  in  the  usual  way,  the  darkest  part  com- 
mencing about  where  the  shade  line  would  actually 
be  on  the  surface,  and  the  lightest  portion  a  little 
to  the  left  of  the  center.  Fig.  81  is  a  cylinder 
showing  the  heaviest  shade  at  the  right,  as  this 
method  is  often  used.  Considerable  practice  is 
necessary  in  order  to  obtain  good  results;  but  in 
this,  as  in  other  portions  of  mechanical  drawing, 


Fig.  80. 


perseverance  has  its  reward.  Fig.  82  represents  a  cylinder  in  a 
horizontal  position,  and  Fig.  83  represents  a  section  of  a  hollow 
vertical  cylinder. 


Fig.  81. 


Fig.  82. 


Fig.  83. 


Figs.  84  to  87  give  other  examples  of  familiar  objects. 

In  the  elevation  of  the  cone  shown  in  Fig.  87  the  shade  lines 
should  diminish  in  weight  as  they  approach  the  apex.  Unless 
this  is  done  it  will  be  difficult  to  avoid  the  formation  of  a  blot  at 
that  point. 


345 


120 


MECHANICAL  DRAWING 


LETTERING. 

All  working  drawings  require  more  or  less  lettering,  such  as 
titles,  dimensions,  explanations,  etc.  In  order  that  the  drawing 
may  appear  finished,  the  lettering  must  be  well  done.  No  style 
of  lettering  should  ever  be  used  which  is  not  perfectly  legible. 
It  is  generally  best  to  use  plain,  easily-made  letters  which  present 


Pig.  84. 


Pig.  85. 


Fig.  86. 


Pig.  87 


a  neat  appearance.  Small  letters  used  on  the  drawing  for  notes  or 
directions  should  be  made  free-hand  with  an  ordinary  writing  pen. 
Two  horizontal  guide  lines  should  be  used  to  limit  the  height  of 
the  letters;  after  a  time,  however,  the  upper  guide  line  may  be 
omitted. 


348 


8 

I 

<  i 

11 


MECHANICAL  DRAWING 


In  the  early  part  of  this  course  the  inclined  Gothic  letter  was 
described,  and  the  alphabet  given.  The  Roman,  Gothic  and  block 
letters  are  perhaps  the  most  used  for  titles.  These  letters,  being 
of  comparatively  large  size,  are  generally  made  mechanically;  that 
is,  drawing  instruments  are  used  in  their  construction.  In  order 
that  the  letters  may  appear  of  the  same  height,  some  of  them, 
owing  to  their  shape,  must  be  made  a  little  higher  than  the  others. 
This  is  the  case  with  the  letters  curved  at  the  top  and  bottom, 
such  as  C,  O,  S,  etc.,  as  shown  somewhat  exaggerated  in 
Fig.  88.  Also,  the  letter  A  should  extend  a  little  above,  and  V  a 
little  below,  the  guide  lines,  because  if  made  of  the  same  height 
as  the  others  they  will  appear  shorter.  This  is  true  of  all  capitals, 
whether  of  Roman,  Gothic,  or  other  alphabets.  In  the  block  letter, 
however,  they  are  frequently  all  of  the  same  size. 

There  is  no  absolute  size  or  proportion  of  letters,  as  the 
dimensions  are  regulated  by  the  amount  of  space  in.  which  the 
letters  are  to  be  placed,  the  size  of  the  drawing,  the  effect  desired, 
etc.  In  some  cases  letters  are  made  so  that  the  height  is  greater 
than  the  width,  and  sometimes  the  reverse;  sometimes  the  height 
and  width  are  the  same. '  This  last  proportion  is  the  most  common. 
Certain  relations  of  width,  however,  should  be  observed.  Thus,  in 
whatever  style  of  alphabet  used,  the  W  should  be  the  widest  letter; 
J  the  narrowest,  M  and  T  next  widest  to  W,  then  A  and  B.  The 
other  letters  are  of  about  the  same  width. 

In  the  vertical  Gothic  alphabet,  the  average  height  is  that  of 
B,  D,  E,  F,  etc.,  and  the  additional  height  of  the  curved  letters 
and  of  the  A  and  V  is  very  slight.  The  horizontal  cross  lines  of 
such  letters  as  E,  F,  H,  etc.,  are  slightly  above  the  center;  those 
of  A,  G  and  P  slightly  below. 

For  the  inclined  letters,  60°  is  a  convenient  angle,  although 
they  may  be  at  any  other  angle  suited  to  the  convenience  or  fancy 
of  the  draftsman.  Many  draftsmen  use  an  angle  of  about  70°. 

The  letters  of  the  Roman  alphabet,  whether  vertical  or 
inclined,  are  quite  ornamental  in  effect  if  well  made,  the  inclined 
Roman  being  a  particularly  attractive  letter,  although  rather 
difficult  to  make.  The  block  letter  is  made  on  the  same  general 
plan  as  the  Gothic,  but  much  heavier.  Small  squares  are  taken  as 


122 


MECHANICAL     DRAWING. 


1 1 

i— 4 


Kf  f>lflp 

i  ^ii  \>\  i 

ij>J  ,04 

!ir 


xl 


x 


a 


00 

Pi: — *     IN^J! 
,   „ 

I       '' 

H 

M 

Pi 
.    o 

UX^i 
•      JDLJ 

IdP!! 


s 


MECHANICAL   DRAWING  12.3 

the  unit  of  measurement,  as  shown.  The  use  of  this  letter  is  not 
advocated  for  general  work,  although  if  made  merely  in  outline  the 
effect  is  pleasing.  The  styles  of  numbers  corresponding  with 
the  alphabets  of  capitals  given  here,  are  also  inserted.  When  a 
fraction,  such  as  2|  is  to  be  made,  the  proportion  should  be  about 
as  shown.  For  small  letters,  usually  called  lower-case  letters, 

abcdefghijklmn 
opqrstuvwxyz 


Fig.  89. 


abcdefgh/jk/mr? 
opqrs 


Fig.  90. 


abcdefghijklmn 
op  qr  s  tuvwxyz 


Fig.  91. 


the  height  may  be  made  about  two-thirds  that  of  the  capitals. 
This  proportion,  however,  varies  in  special  cases. 

The  principal  lower-case  letters  in  general  use  among  drafts- 
men are  shown  in  Figs.  89,  90,  91  and  92.  The  Gothic  letters 
shown  in  Figs.  89  and  90  are  much  easier  to  make  than  the 
Roman  letters  in  Figs.  91  and  92.  These  letters,  however,  do  not 


124 


MECHANICAL     DRAWING. 


350 


MECHANICAL   DRAWING 


give  as  finished  an  appearance  as  the  Roman.  As  has  already 
been  stated  in  Mechanical  Drawing,  Part  I,  the  inclined  letter  is 
easier  to  make  because  slight  errors  are  not  so  apparent. 

One  of  the  most  important  points  to  be  remembered  in  letter- 
ing is  the  spacing.  If  the  letters  are  finely  executed  but  poorly 
spaced,  the  effect  is  not  good.  To  space  letters  correctly  and 
rapidly,  requires  considerable  experience;  and  rules  are  of  little 
value  on  account  of  the  many  combinations  in  which  letters  are 

abcdefghijkLmn 
opqrstuvwxyz 

Fig.  92. 

found.  A  few  directions,  however,  may  be  found  helpful.  For 
instance,  take,  the  word  TECHNICALITY,  Fig.  93.  If  all  the 
spaces  were  made  equal,  the  space  between  the  L  and  the  I  would 
appear  to  be  too  great,  and  the  same  would  apply  to  the  space 
between  the  I  and  the  T.  The  space  between  the  H  and  the  N 
and  that  between  the  N  and  the  I  would  be  insufficient.  In 
general,  when  the  vertical  side  of  one  letter  is  followed  by  the  verti- 
cal side  of  another,  as  in  H  E,  H  B,  I  R,  etc.,  the  maximum  space 

TECHNICALITY 

Fig.  93. 

should  be  allowed.  Where  T  and  A  come  together  the  least  space 
is  given,  for  in  this  case  the  top  of  the  T  frequently  extends  over 
the  bottom  of  the  A.  In  general,  the  spacing  should  be  such  that 
a  uniform  appearance  is  obtained.  For  the  distances  between 
words  in  a  sentence,  a  space  of  about  \\  the  width  of  the  average 
letter  may  be  used.  The  space,  however,  depends  largely  upor 
desired  effect 


351 


126 


MECHANICAL   DRAWING 


For  large  titles,  such  as  those  placed  on  charts,  maps,  and 
some  large  working  drawings,  the  letters  should  be  penciled  before 
inking.  If  the  height  is  made  equal  to  the  width  considerable 
time  and  labor  will  be  saved  in  laying  out  the  work.  This  is 
especially  true  with  such  Gothic  letters  as  O,  Q,  C,  etc.,  as  these 
letters  may  then  be  made  with  compasses.  If  the  letters  are  of 
sufficient  size,  the  outlines  may  be  drawn  with  the  ruling  pen  or 
compasses,  and  the  spaces  between  filled  in  with  a  fine  brush. 

The  titles  for  working  drawings  are  generally  placed  in  the 
lower  right-hand  corner.  Usual  a  draftsman  has  his  choice  of 


•B 


Block  Letters. 

letters,  mainly  because  after  he  has  become  used  to  making  one 
style  he  can  do  it  rapidly  and  accurately.  However,  in  some  draft-, 
ing  rooms  the  head  draftsman  decides  what  lettering  shall  be  used. 
In  making  these  titles,  the  different  alphabets  are  selected  to  give 
the  best  results  without  spending  too  much  time.  In  most  work 
the  letters  are  made  in  straight  lines,  although  we  frequently  find 
a  portion  of  the  title  lettered  on  an  arc  of  a  circle. 

In  Fig.  94  is  shown  a  title  having  the  words  CONNECTING 
ROD  lettered  on  an  arc  of  a  circle.  To  do  this  work  requires 
considerable  patience  and  practice.  First  draw  the  vertical  center 


352 


MECHANICAL   DRAWING  127 


line  as  shown  at  C  in  Fig.  94.  Then  draw  horizontal  lines  for  the 
horizontal  letters.  The  radii  of  the  arcs  depend  upon  the  general 
arrangement  of  the  entire  title,  and  this  is  a  matter  of  taste.  The 
difference  between  the  arcs  should  equal  the  height  of  the  letters. 
After  the  arc  is  drawn,  the  letters  should  be  sketched  in  pencil  to 
find  their  approximate  positions.  After  this  is  done,  draw  radial 
lines  from  the  center  of  the  letters  to  the  center  of  the  arcs. 


X 


/\J  FOR 

BEAM  ENGINE     ' 

SCALE  3  INCHES   ==   1   FOOT 

PORTLAND    COMPANY'S  WORKS 

JULY  1O,   1894. 

Pig.  04. 


These  lines  will  be  the  centers  of  the  letters,  as  shown  at  A,  B,  D 
and  E.  The  vertical  lines  of  the  letters  should  not  radiate  from 
the  center  of  the  arc,  but  should  be  parallel  to  the  center  lines 
already  drawn;  otherwise  the  letters  will  appear  distorted.  Thus, 
in  the  letter  N  the  two  verticals  are  parallel  to  the  line  A. 
same  applies  to  the  other  letters  in  the  alphabet 


353 


128  MECHANICAL  DRAWING 

Tracing.  Having  finished  the  pencil  drawing,  the  next  ctep 
is  the  inking.  In  some  offices  the  pencil  drawing  is  made  on  a  thin, 
tough  paper,  called  board  paper,  and  the  inking  is  done  over  the 
pencil  drawing,  in  the  manner  with  which  the  student  is  already 
familiar.  It  is  more  common  to  do  the  inking  on  thin,  trans, 
parent  cloth,  called  tracing  cloth,  which  is  prepared  for  the  pur- 
pose. This  tracing  cloth  is  made  of  various  kinds,  the  kind  in 
ordinary  use  being  what  is  known  as  "  dull  back,"  that  is,  one 
side  is  finished  and  the  other  side  is  left  dull.  Either  side  may 
be  used  to  draw  upon,  but  most  draftsmen  prefer  the  dull  side. 
If  a  drawing  is  to  be  traced  it  is  a  good  plan  to  use  a  311  or  4H 
pencil,  so  that  the  lines  may  be  easily  seen  through  the  cloth. 

The  tracing  cloth  is  stretched  smoothly  over  the  pencil  draw- 
ing  and  a  little  powdered  chalk  rubbed  over  it  with  a  dry  cloth, 
to  remove  the  slight  amount  of  grease  or  oil  from  the  surface  and 
make  it  take  the  ink  better.  The  dust  must  be  carefully  brushed 
or  wiped  off  with  a  soft  cloth,  after  the  rubbing,  or  it  will  inter- 
fere with  the  inking. 

The  drawing  is  then  made  in  ink  on  the  tracing  cloth,  after 
the  same  general  rules  as  for  inking  the  paper,  but  care  must  be 
taken  to  draw  the  ink  lines  exactly  over  the  pencil  lines  which 
are  on  the  paper  underneath,  and  which  should  be  just  heavy 
enough  to  be  easily  seen  through  the  tracing  cloth.  The  ink  lines 
should  be  firm  and  fully  as  heavy  as  for  ordinary  work.  In  tracing, 
it  is  better  to  complete  one  view  at  a  time,  because  if  parts  of 
several  views  are  traced  and  the  drawing  left  for  a  day  or  two,  the 
cloth  is  liable  to  stretch  and  warp  so  that  it  will  be  difficult  to 
complete  the  views  and  make  the  new  lines  fit  those  already 
drawn  and  at  the  same  time  conform  to  the  pencil  lines  under- 
neath. For  this  reason  it  is  well,  when  possible,  to  complete  a 
view  before  leaving  the  drawing  for  any  length  cf  time,  although 
of  course  on  viewc  in  which  there  is  a  good  deal  of  work  this 
cannot  always  be  done.  In  this  case  the  draftsman  must  manipu- 
late his  tracing  cloth  and  instruments  to  make  the  lines  fit  as  best 
he  can.  A  skillful  draftsman  will  have  no  trouble  from  this 
source,  but  the  beginner  may  at  first  find  difficulty. 

Inking  on  tracing  cloth  will  be  found  by  the  beginner  to  be 
quite  different  from  inking  on  the  paper  to  which  he  has  been 
accustomed,  and  he  will  doubtless  make  many  blots  and  think  ai 


354 


MECHANICAL  DRAWING  120 

first  that  it  is  hard  to  make  a  tracing.  After  a  little  practice, 
nowever,  he  will  find  that  the  tracing  cloth  is  very  satisfactory 
and  that  a  good  drawing  can  be  made  on  it  quite  as  easily  as  on 
paper. 

The  necessity  for  making  erasures  should  be  avoided,  as  far 
as  possible,  but  when  an  erasure  must  be  made  a  good  ink  rubber 
or  typewriter  eraser  may  be  used.  If  the  erased  line  is  to  have 
ink  placed  on  it,  such  as  a  line  crossing,  it  is  better  to  use  a  soft 
rubber  eraser.  All  moisture  should  be  kept  from  the  cloth. 

Blue  Printing,  The  tracing,  of  course,  cannot  be  Bent  into 
the  shop  for  the  workmen  to  use,  as  it  would  soon  become  soiled 
and  in  time  destroyed,  so  that  it  is  necessary  to  have  some  cheap 
and  rapid  means  of  making  copies  from  it.  These  copies  are 
made  by  the  process  of  blue  printing  in  which  the  tracing  is  used 
in  a  manner  similar  to  the  use  made  of  a  negative  in  photography. 

Almost  all  drafting  rooms  have  a  frame  for  the  purpose  of 
making  blue  prints.  These  frames  are  made  in  many  styles,  some 
simple,  some  elaborate.  A  simple  and  efficient  form  is  a  flat  sur- 
face usually  of  wood,  covered  with  padding  of  soft  material,  such 
as  felting.  To  this  is  hinged  the  cover,  which  consists  of  a  frame 
similar  to  a  picture  frame,  in  which  is  set  a  piece  of  clear  glass. 
The  whole  is  either  mounted  on  a  track  or  on  some  sort  of  a 
swinging  arm,  so  that  it  may  readily  be  run  in  and  out  of  a 
window. 

The  print  is  made  on  paper  prepared  for  the  purpose  by 
having  one  of  its  surfaces  coated  with  chemicals  which  are  sensi- 
tive to  sunlight.  This  coated  paper,  or  blue-print  paper,  as  it  is 
called,  is  laid  on  the  padded  surface  of  the  frame  with  its  coated 
side  uppermost;  the  tracing  laid  over  it  right  side  up,  and  the 
glass  pressed  down  firmly  and  fastened  in  place.  Springs  are 
frequently  used  to  keep  the  paper,  tracing,  etc.,  against  the  glass. 
With  some  frames  it  is  more  convenient  to  turn  them  over  and 
remove  the  backs.  In  such  cases  the  tracing  is  laid  against  the 
glass,  face  down;  the  coated  paper  is  then  placed  on  it  with  the 
coated  side  against  the  tracing  cloth. 

The  sun  is  allowed  to  shine  upon  the  drawing  for  a  few 
minutes,  then  the  blue-print  paper  is  taken  out  and  thoroughly 
washed  in  clean  water  for  several  minutes  and'  hung  up  to  dry. 


355 


130  MECHANICAL  DRAWING 

If  the  paper  has  been  recently  prepared  and  the  exposure  properly 
timed,  the  coated  surface  of  the  paper  will  now  be  of  a  clear,  deep 
blue  color,  except  where  it  was  covered  by  the  ink  lines,  where  it 
will  be  perfectly  white. 

The  action  has  been  this:  Before  the  paper  was  exposed  to 
the  light  the  coating  was  of  a  pale  yellow  color,  and  if  it  had  then 
been  put  in  water  the  coating,  would  have  all  washed  off,  leaving 
the  paper  white.  In  other  words,  before  being  exposed  to  the 
sunlight  the  coating  was  soluble.  The  light  penetrated  the  trans- 
parent tracing  cloth  and  acted  upon  the  chemicals  of  the  coating, 
changing  their  nature  so  that  they  became  insoluble;  that  is,  when 
put  in  water,  the  coating,  instead  of  being  washed  off,  merely 
turned  blue.  The  light  could  not  penetrate  the  ink  with  which 
the  lines,  figures,  etc.,  were  drawn,  consequently  the  coating  under 
these  was  not  acted  upon  and  it  washed  off  when  put  in  water, 
leaving  a  white  copy  of  the  ink  drawing  on  a  blue  background. 
If  running  water  cannot  be  used,  the  paper  must  be  washed  in  a 
sufficient  number  of  changes  until  the  water  is  clear.  It  is  a  good 
plan  to  arrange  a  tank  having  an  overflow,  so  that  the  water  may 
remain  at  a  depth  of  about  0  or  8  inches. 

The  length  of  time  to  which  a  print  should  be  exposed  to  the 
light  depends  upon  the  quality  and  freshness  of  the  paper,  the 
chemicals  used  and  the  brightness  of  the  light.  Some  paper  is 
prepared  so  that  an  exposure  of  one  minute,  or  even  less,  in  bright 
sunlight,  will  give  a  good  print  and  the  time  ranges  from  this  to 
twenty  minutes  or  more,  according  to  the  proportions  of  ths 
various  chemicals  in  the  coating.  If  the  full  strength  of  the  sun- 
light  does  not  strike  the  paper,  as,  for  instance,  if  clouds  partly 
cover  the  sun,  the  time  of  exposure  must  be  lengthened. 

Assembly  Drawing.  We  have  followed  through  the  process 
of  making  a  detail  drawing  from  the  sketches  to  the  blue  print 
ready  for  the  workmen.  Such  a  detail  drawing  or  set  of  drawings 
shows  the  form  and  size  of  each  piece,  but  does  not  show  how  the 
pieces  go  together  and  gives  no  idea  of  the  machine  as  a  whole. 
Consequently,  a  general  drawing  or  assembly  drawing  must  be 
made,  which  will  show  these  things.  Usually  two  or  more  views 
are  necessary,  the  number  depending  upon  the  complexity  of  the 
machine.  Very  often  a  cross-section  through  some  part  of  the 


MECHANICAL  DRAWING  131 

machine,  chosen  so  as  to  give  the  best  general  idea  with  the  least 
amount  of  work,  will  make  the  drawing  clearer. 

The  number  of  dimensions  required  on  an  assembly  drawing 
depends  largely  upon  the  kind  of  machine.  It  is  usually  best  to 
give  the  important  over-all  dimensions  and  the  distance  between 
the  principal  center  lines.  Care  must  be  taken  that  the  over-all 
dimensions  agree  with  the  sum  of  the  dimensions  of  the  various 
details.  For  example,  suppose  three  pieces  are  bolted  together, 
the  thickness  of  the  pieces  according  to  the  detail  drawing,  being 
one  inch,  two  inches,  and  five  and  one-half  inches  respectively;  the 
sum  of  these  three  dimensions  is  eight  and  one-half  inches  and 
the  dimensions  from  outside  on  the  assembly  drawing,  if  given  at 
all,  must  agree  with  this.  It  is  a  good  plan  to  add  these  over-all 
dimensions,  as  it  serves  as  a  check  and  relieves  the  mechanic  of  the 
necessity  of  adding  fractions. 

FORMULA  FOR  BLUE=PRINT  SOLUTION. 

Dissolve  thoroughly  and  filter. 

Red  Prussiate  of  potash 2^  ounces, 

A'    Water 1     pint. 

Ammonio-Citrate  of  iron 4  ounces, 

B-     Water 1  Pint 

Use  equal  parts  of  A  and  B. 

FORnULA  FOR  BLACK  PRINTS 
Negatives.     White  lines  on  blue  ground;  prepare  the  paper 

with 

Ammonio-Citrate  of  iron 40  grains, 

Water !  ounce' 

After  printing  wash  in  water. 

Positives.     Black  lines  on  white  ground;  prepare  the  paper 

Iron  perchloride 616  grains, 

Oxalic  Acid 308  grains, 

Water Hounces. 

(Gallic  Acid 1   ounce» 

Develop  in  ^Citric  Acid l  ounce» 

(Alum 8ounce& 

Use  1J  ounces  of  developer  to  one  gallon  of  water.     Paper  is 
fully  exposed  when  it  has  changed  from  yellow  to  white. 


357 


132  MECHANICAL  DRAWING 

PLATES. 

PLATE  IX. 

The  plates  of  this  Instruction  Paper  should  be  laid  out  at  the 
same  size  as  the  plates  in  Parts  I  and  II.  The  center  lines  and 
border  lines  should  also  be  drawn  as  described. 

First  draw  two  ground  lines  across  the  sheet,  3  inches  below 
the  upper  border  line  and  3  inches  above  the  lower  border  line. 
The  first  problem  on  each  ground  line  is  to  be  placed  1  inch  from 
the  left  border  line;  and  spaces  of  about  1  inch  should  be  left 
between  tbe  figures. 

Isolated  points  are  indicated  by  a  small  cross  X,  and  projections 
of  lines  are  to  be  drawn  full  unless  invisible.  All  construction 
lines  should  be  fine  dotted  lines.  Given  and  required  lines  should 
be  drawn  full. 

Problems  on  Upper  Ground  Line: 

1.  Locate  both  projections  of  a  point  on  the  horizontal  plane 
1  inch  from  the  vertical  plane. 

2.  Draw  the  projections  of  a  line  2  inches  long  which  is 
parallel  to  the  vertical  plane  and  which  makes  an  angle  of  45 
degrees  with  the  horizontal  plane  and  slants  upward  to  the  right. 

The  line  should  be  1  inch  from  the  vertical  plane  and  the  lower  end 
%  inch  above  the  horizontal. 

3  Draw  the  projections  of  a  line  1£  inches  long  which  is 
parallel  to  both  planes.  1  inch  above  the  horizontal,  and  |  inch  from 
the  vertical. 

4.  Draw  the  plan  and  elevation  of  a  line  2  incnes  long  which 
is  parallel  to  H  and  makes  an  angle  of  30  degrees  with  V.     Let  the 
right-hand  end  of  the  line  be  the  end  nearer  V,  ^  inch  from  V. 
The  line  to  be  1  inch  above  H. 

5.  Draw  the  plan  and  elevation  of  a  line  1£  inches  long 
which  is  perpendicular  to  the  horizon  till  plane  and  1  inch  from  the 
vertical.    Lower  end  of  line  is  ^  inch  above  H. 

6.  Draw  the  projections  of  a  line  1  inch  long  which   is 
perpendicular    to  the  vertical  plane   and  1£  inches  above    the 
horizontal.    The  end  of  the  lino  nearer  V,  or  the  back  end,  is 

inch  from  V. 


358 


X 


( X 


t 


MECHANICAL  DRAWING  133 

7.  Draw  two  projections  which  shall  represent  a  line  oblique 
to  both  planes. 

NOTE.    Leave  1  inch  between  this  figure  and  the  right-hand  border  line. 
Problems  on  Lower  Ground  Line: 

8.  Draw  the  projections  of  two  parallel  lines  each  1^  inches 
long.    The  lines  are  to  be  parallel  to  the  vertical  plane  and  to  make 
angles  of  60  degrees  with  the  horizontal.    The  lower  end  of  each 
lino  is  \  inch  above  H.    The  right-hand  end  of  the  right-hand  line 
is  to  be  2|  inches  from  the  left-hand  margin. 

9.  Draw  the  projections  of  two  parallel  lines  each  2  inches 
long.    Both  lines  to  be  parallel  to  the  horizontal  and  to  make 
an  angle  of  30  degrees  with  the  vertical.    The  lower  line  to  be 
|  inch  above  H,  and  one  end  of  one  line  to  be  against  V. 

10.  Draw  the  projections  of  two  intersecting  lines.     One 
2  inches  long  to  be  parallel  to  both  planes,  1  inch  above  H,  and 
|  inch  from  the  vertical;  and  the  other  to  be  oblique  to  both 
planes  and  of  any  desired  length. 

11.  Draw  plan  and  elevation  of  a  prism  1  inch  square  and  1^ 
inches  long.    The  prism  to  have  one  side  on  the  horizontal  plane, 
and  its  long  edges  to  be  perpendicular  to  V.    The  back  end  of  the 
prism  is  \  inch  from  the  vertical  plane. 

12.  Draw  plan  and  elevation  of  a  prism  the  same  size  as  given 
above,  but  with  the  long  edges  parallel  to  both  planes,  the  lower 
face  of  the  prism  to  be  parallel  to  H  and  \  inch  above  it,    The 
back  face  to  be  \  inch  from  V. 

PLATE  X. 

The  ground  line  is  to  be  in  the  middle  of  the  sheet,  and  the 
location  and  dimensions  of  the  figures  are  to  be  as  given,  The 
Qrst  figure  shows  a  rectangular  block  with  a  rectangular  hole  cut 
through  from  front  to  back.  The  other  two  figures  represent  the 
same  block  in  different  positions.  The  second  figure  is  the  end  or 
profile  projection  of  the  block.  The  same  face  is  on  H  in  all 
three  positions.  Be  careful  not  to  omit  the  shade  lines.  The 
figures  given  on  the  plate  for  dimensions,  etc.,  are  to  be  used  but 
not  repeated  on  the  plate  by  the  stuient. 


134  MECHANICAL  DRAWING 

PLATE  XI. 

Three  ground  lines  are  to  be  used  on  this  plate,  two  at  the  left 
4£  inches  long  and  3  inches  from  top  and  bottom  margin  lines;  and 
one  at  the  right,  half  way  between  the  top  and  bottom  margins,  9£ 
inches  long. 

The  figures  1,  2,  3  and  4  are  examples  for  finding  the  true 
lengths  of  the  lines.  Begin  No.  1  finch  from  the  border,  the 
vertical  projection  If  inches  long,  one  end  on  the  ground  line  and 
inclined  at  30°.  The  horizontal  projection  has  one  end  \  incl 
from  V,  and  the  other  \\  inches  from  V.  Find  the  true  length  of 
the  line  by  completing  the  construction  commenced  by  swinging 
the  arc,  as  shown  in  the  figure. 

Locate  the  left-hand  end  of  No.  2  3  inches  from  the  border, 
1  inch  above  H,  and  |  inch  from  V.  Extend  the  vertical  projection 
to  the  ground  line  at  an  angle  of  45°,  and  make  the  horizontal  pro- 
jection at  30°.  Complete  the  construction  for  true  length  as 
commenced  in  the  figure. 

In  Figs.  3  and  4,  the  true  lengths  are  to  be  found  by  complet- 
ing the  revolutions  indicated.  The  left-hand  end  of  Fig.  3  is  | 
inch  from  the  margin,  \\  inches  from  V,  and  1|  inches  above  H. 
The  horizontal  projection  makes  an  angle  of  60°  and  extends  to  the 
ground  line,  and  the  vertical  projection  is  inclined  at  45°. 

The  fourth  figure  is  3  inches  from  the  border,  and  represents 
a  line  in  a  profile  plane  connecting  points  a  and  1.  a  is  1 J  inches 
above  H  and  £  inch  from  V;  and  I  is  £  inch  above  H  and  1^ 
inches  from  V. 

The  figures  for  the  middle  ground  line  represent  a  pentagonal 
pyramid  in  three  positions.  The  first  position  is  the  pyramid  with 
the  axis  vertical,  and  the  base  f  inch  above  the  horizontal.  The 
height  of  the  pyramid  is  2^  inches,  and  the  diameter  of  the  circle 
circumscribed  about  the  base  is  2$  inches.  The  center  of  the  circle 
is  6  inches  from  the  left  margin  and  If  inches  from  V.  Spaces 
between  figures  to  be  f  inch. 

In  the  second  figure  the  pyramid  has  been  revolved  about  the 
right-hand  corner  of  the  base  as  an  axis,  through  an  angle  of  15°. 
The  axis  of  the  pyramid,  shown  dotted,  is  therefore  at  75°.  The 
method  of  obtaining  75°  and  15°  with  the  triangles  was  shown  in 


364 


MECHANICAL   DKAWING 


Part  I.  From  the  way  in  which  the  pyramid  has  been  revolved, 
all  angles  with  V  must  remain  the  same  as  in  the  first  position, 
hence  the  vertical  projection  will  be  the  same  shape  and  size  as 
before.  All  points  of  the  pyramid  remain  the  same  distance 
from  V.  The  points  on  the  plan  are  found  on  T-square  lines 
through  the  corners  of  the  first  plan  and  directly  beneath  the 
points  in  elevation.  In  the  third  position  the  pyramid  has  been 
swung  around,  about  a  vertical  line  through  the  apex  as  axis, 
through  30°.  The  angle  with  the  horizontal  plane  remains  the 
same;  consequently  the  plan  is  the  same  size  and  shape  as  in  the 


Fig.  96. 

second  position,  but  at  a  different  angle  with  the  ground  line. 
Heights  of  all  points  of  the  pyramid  have  not  changed  this  time, 
and  hence  are  projected  across  from  the  second  elevation.  Shade 
lines  are  to  be  put  on  between  the  light  and  dark  surfaces  as 
determined  by  the  45°  triangle. 

PLATE  XII. 
Developments. 

On  this  plate  draw  the  developments  of  a  truncated  octagonal 
prism,  and  of  a  truncated  pyramid  having  a  square  base.  The 
arrangement  on  the  plate  is  left  to  the  student;  but  we  should 
suggest  that  the  truncated  prism  and  its  development  be  place 


136 


MECHANICAL   DKAW1NG 


the  left,  and  that  the  development  of  the  truncated  pyramid  be 
placed  under  the  development  of  the  prism;  the  truncated  pyramid 
may  be  placed  at  the  right. 

The  prism  and  its  development  are  shown  in  Fig.  96.  The 
prism  is  3  inches  high,  and  the  base  is  inscribed  in  a  circle  2£ 
inches  in  diameter.  The  plane  forming  the  truncated  prism  is 
passed  as  indicated,  the  distance  A  B  being  1  inch.  Ink  a  suffi- 
cient number  of  construction  lines  to  show  clearly  the  method  of 
finding  the  development. 

The  pyramid  and  its  development  are  shown  in  Fig.  97.  Each 
side  of  the  square  base  is  2  inches,  and  the  altitude  is  3£  inches. 
A 


Pig.  97. 

The  plane  forming  the  truncated  pyramid  is  passed  in  such  a 
position  that  A  B  equals  If  inches,  and  A  C  equals  2£  inches.  In 
this  figure  the  development  may  be  drawn  in  any  convenient 
position,  but  in  the  case  of  the  prism  it  is  better  to  draw  the 
development  as  shown.  Indicate  clearly  the  construction  by 
inking  the  construction  lines. 

PLATE  XIII. 

Isometric  and  Oblique  Projection. 

Draw  the  oblique  projection  of  a  portable  closet.  The  angle  to 
be  used  is  45°.  Make  the  height  3£-  inches,  the  depth  1£  inches, 
and  the  width  3  inches.  See  Fig.  98.  The  width  of  the  closet 


868 


MECHANICAL   DRAWING 


137 


- 


r      -  -i 



.. 

—  1 

*-,£- 

= 

C 

tvi 

? 

°* 

^ 

C 

j 

£ 

^ 

. 

i  

j 

—  i" 

ri  —  » 

3 

i 

Fig.  98. 

centrally  in  the  front  of  the  closet,  the  bottom  edge  at  the  height 
of  the  floor  of  the  closet,  the  hinges  of  the  door  to  be  placed  on  the 
left-hand  side.  In  the  oblique  drawing,  show  the  door  opened 
at  an  angle  of  90  degrees.  The  thickness  of  the  material  of  the 
closet,  door,  and  floor  is  ^  inch. 
The  door  should  be  hung  so  that 
when  closed  it  will  be  flush  with 
the  front  of  the  closet. 

Make  the  isometric  drawing 
of  the  flight  of  steps  and  end  walls 
as  shown  by  the  end  view  in  Fig. 
99.  The  lower  right-hand  corner 
is  to  be  located  2£  inches  from 
the  lower,  and  5  inches  from  the  Fig-  "• 

right-hand,  margin.  The  base  of  the  end  wall  is  3£  inches  Jong, 
and  the  height  is  2£  inches.  Beginning  from  the  back  of  the 
wall,  the  top  is  horizontal  for  f  inch,  the  remainder  of  the  outline 
being  comix>sed  of  arcs  of  circles  whose  radii  and  centers  are  given 


138  MECHANICAL   DRAWING 

in  the  figure.  The  thickness  of  the  end  wall  is  f  inch,  and  both 
ends  are  alike.  There  are  to  be  five  steps;  each  rise  is  to  be 
|  inch,  and  each  tread  \  inch,  except  that  of  the  top  step,  which 
is  I  inch.  The  first  step  is  located  f  inch  back  from  the  corner 
of  the  wall.  The  end  view  of  the  wall  should  be  constructed  on  a 
separate  sheet  of  paper,  from  the  dimensions  given,  the  points  on 
the  curve  being  located  by  horizontal  co-ordinates  from  the  vertical 
edge  of  the  wall,  and  then  these  co-ordinates  transferred  to  the 
isometric  drawing.  After  the  isometric  of  one  curved  edge  has 
been  made,  the  others  can  be  readily  found  from  this.  The  width 
of  the  steps  inside  the  walls  is  3  inches. 

PLATE  XIV. 

Free-hand  Lettering. 

On  account  of  the  importance  of  free-hand  lettering,  the 
student  should  practice  it  at  every  opportunity.  For  additional 
practice,  and  to  show  the  improvement  made  since  completing 
Part  I,  lay  out  Plate  XIV  in  the  same  manner  as  Plate  I,  and  letter 
all  four  rectangles.  Use  the  same  letters  and  words  as  in  the  lower 
light-hand  rectangle  of  Plate  I. 

PLATE  XV. 

Lettering. 

First  lay  out  Plate  XV  in  the  same  manner  as  previous 
plates.  After  drawing  the  vertical  center  line,  draw  light  pencil 
lines  as  guide  lines  for  the  letters.  The  height  of  each  line  of 
letters  is  shown  on  the  reproduced  plate.  The  distance  be- 
tween the  letters  should  be  ^  inch  in  every  case.  The  spacing 
of  the  letters  is  left  to  the  student.  He  may  facilitate  his  work 
by  lettering  the  words  on  a  separate  piece  of  paper,  and  finding 
the  center  by  measurement  or  by  doubling  the  paper  into  two 
equal  parts.  The  styles  of  letters  shown  on  the  reproduced  plate 
should  be  used 


374 


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REVIEW  QUESTIONS. 


PRACTICAL  TEST  QUESTIONS. 

In  the  foregoing  sections,  of  this  Cyclopedia 
numerous  illustrative  examples  are  worked  out  in 
ill  m  order  to  show  the  application  of  the  various 
methods  and  principles.  Accompanying  these  are 
examples  for  practice  which  will  aid  the  reader  in 
fixing  the  principles  in  mind. 

In  the  following  pages  are  given  a  large  number 
of  test  questions  and  problems  which  afford  a  valu- 
able means  of  testing  the  reader's  knowledge  of  the 
subjects  treated.  They  will  be  found  excellent  prac- 
tice for  those  preparing  for  College,  Civil  Service,  or 
Engineer's  License.  In  some  cases  numerical  answers 
are  given  as  a  further  aid  in  this  work. 


REVIEW    QUESTIONS 

ON      THE      SCJBJKCT      OF 

PLANK    SURVEYING 


1.  Explain  by    a  diagram  how    to  erect    (with  the  tape 
alone)  a  line  at  right  angles  to  a  given  line. 

2.  The  sides  of  a  triangular  field  are  820,432  and  529  feet. 
Find  the  area  of  the  field  in  acres,  rods  and  square  rods. 

3.  Find  the  area  of  a  triangle  \vhose  sides  are  31,  40  and 
55  rods. 

4.  Given  in  Fig.  14,  C  B  =   2.85  chains,  C  D  =  3.67 
chains,  C  S  =  C  L  =  0.52  chains  and  L  S  =  0.75  chains.     Cal- 
culate the    area  of  the  triangle  B  C  D. 

5.  A  certain  line  is  known  to  be  530  feet  in  length,  but 
when  measured  with  a  certain  tape  is  found  to  be  533i  feet  in 
length.     Determine  the  true  length  of  the  tape. 

6.  A  certain  field  is  measured  with  a  Gunter's  chain  and 
is  found  to  contain  5.75  acres.     It  is  afterwards  discovered  that  the 
chain  is  ^   of  a  foot  too  long.     Find  the  true  area  of  the  field. 

7.  If  a  line  as  measured  is  found  to  be  432|  feet  in  length 
and  it  is  afterwards  discovered  that  the  tape  is  too  short  by  |  of  a 
foot,  what  is  the  true  length  of  the  line? 

8.  A  level  bubble  has  a  radius  of  150  feet  and  its  scale 
has  10  spaces  in  an  inch.     What  error  in  leveling  will  result  at  a 
distance  of  275  feet  when  the  level  bubble  is  li  spaces  out  of  level? 

9.  At  a  distance  of  150  feet,  two  rod  readings  were  3.704 
and    3.745  and   the  bubble  moved  over  1  inch.     Determine  the 
radius  of  the  bubble  tube. 

10.  What  error  in  leveling  will  result  at  a  distance  of  123 
feet  if  the  bubble  is  2|  spaces  out  of  level,  the  scale  of  which 
has  7  spaces  in  an  inch,  the  radius  being  176  feet? 


PLANE  SURVEYING 


11.  If  the  difference  of  rod  readings  is  0.03  foot  and  the 
bubble  having  a  radius  of  50  feet  moves  over  0.015  foot,  deter- 
mine the  distance  from  the  instrument  to  the  rod. 

12.  Given  the  distances  measured  along  the  straight  line 
A  B  in  Fig.  A.  Page  22,  to  be  35,  40,  55,45,  50,  70,  30,  45  and  25 
feet.     The  offsets  beginning  with  0  at  A  are  40,  35, 0,-55,  -65,  -30, 
0,  25  and  0  feet  at  B.     Determine    the  area  included  between  the 
line  A  B  and  the  broken  line  A  C  D  E  F  B. 

1'3.        An  angle  is  5  degrees  and  1  minute  and  the  distance 
823  feet.    What  is  the  length  of  the  subtended  arc? 

14.  In  your  own  words,  describe  the  Gunter's  chain,  and 
its  advantages  in  land  surveying.     Describe  the  engineer's  chain 
and  state  what  errors  are  liable  to  occur  in  using  either  chain  for 
measuring  lines. 

15.  In  your  own  words,  descril>e  the  differeut  kinds  of 
tapes  and  explain  fully  the  advantages  of  the  band  tape  over  the 
chain  for  measuring  the  lengths  of  lines. 

10.        In  your  own  words,  describe  the  wye  level,  accompany- 
ing the  description  with  sketches  whenever  necessary. 

17.  Define  the  following  terms:  Line  of  collimation;  instrn- 

O 

mental  parallax;  spherical  aberration;  chromatic  aberration. 

18.  In   your  own  words,  state  the  adjustments  of  the  wye 
level  ///  tJu'u'  ordt-r. 

1(.>.       In  your  own  words,  describe  hew  to  make  the  tests  for 
the  several  adjustments  of  the  wye  level. 

20.  In   your  own  words,  describe  with  sketches,  how  to 
make  several  adjustments  of  the  wye  level,  in  tJieir  ord<>r. 

21.  Define  the  following  terms:    Back-sight;    fore-sight; 
height  of  instrument;  datum  plain;  bench-mark;  p^g;  elevation. 

22.  In  your  own  words,  descril>e  how     to  "set"  the  wye 
level  and  the  field  operations  of  leveling. 

23.  Assuming  your  own  rod  readings,  prepare  a  system  of 
level  notes;  calculating  heights  of  instrument  and  elevations. 

24.  In  your  own  words,  describe  the  Dumpy  level  and  ex- 
plain  how  it  differs  from  the  wye  level. 

25.  State  and  explain  fully  the  adjustments  of  the  Dumpy 
level  ///  their  order. 


REVIEW     QUESTIONS 


ON      THE      S   IT  B  .1  K  O  T      O  V 


PLANE    SURVEYING 


•ART     II 


1.  Give  a  detailed  description  of  the  transit. 

2.  Give  the  reasons  for  each  of  the  adjustments  of  the  transit. 
Draw  a  diagram  in  each  case. 

3.  Describe  fully  in  their  order  how  to  test  the  adjustments  of 
the  transit. 

4.  Describe   fully  in   their  order  how   to  make   the  several 
adjustments  of  the  transit. 

5.  Define  the  following:     Latitude  of  a  station;  longitude  of 
a  line;  double  longitude;  departure. 

6.  Explain  the  use  of  double  longitudes.     Draw  a  figure,  and 
explain  how  latitudes  and  departures  may  be  gotten  from  the  length 
and  bearing  of  a  line. 

7.  In  order  to  find  the  difference  in  height  of  two  peaks  M 
and  N,  a  base  line  A  B  was  laid  off  5,000  feet  long;  and  the  hori- 
zontal  angles   BAM  =  120°  30',  BAN  =  49°  15',   ABM  =  40°  35', 
and  ABN  =  95°  07',  were  read.     At  A,  the  angle  of  elevation  of 
M  was  17°  19',  and  the  angle  of  elevation  of  N  was  18°  45'.     Com- 
pute the  difference  in  the  height  of  the  two  peaks. 

Ans.  K.  is  226.59  feet  above  N. 

8.  Draw  a  figure,  and  deduce  a  general  rule  for  the  double 
longitude  of  any  course  in  terms  of  thedouble  longitude  anddeparture 
of  the  preceding  course. 

9.  Explain  what  is  meant  by  balancing  a  survey. 

10.  Describe  fully  how  the  latitudes  and  departures  of  a  series 
of  courses  may  be  balanced,  and  how  to  determine  the  balanced 
length  and  bearing  of  each  course. 


PLANE  SURVEYING 


11.  Let    c  +  /  =  0.87    feet,   *'  =  4.65   feet,   and  /  -  3°  32'. 
Compute  the  horizontal  distance  to  the  rod  and  the  difference  in 
elevation.     What  error  results  when  c  +  /  is  neglected? 

12.  In  order  to  find  the  direction  and  distance  between  two 
points  R  and  Q,  the  following  lines  are  run.        RA,  S  87°  37',  W. 
930.57  feet ;  J  J5,  W  621.03  feet;  BQ,  S  88°  15'  West,  82.78  feet. 
Compute   the    bearing   and    length    of   RQ,  and   locate  the  point 
where  it  crosses  A  B,  with  reference  to  A. 

13.  In  order  to  find  the  direction  and  distance  of  a  point,  the 
following    lines    are    run:    AC  X  42°  15'  E  714.5  feet.     CB  X  1°  8' 
E  210.5  feet.     Compute  the  distance  and  bearing  of  the  two  points. 

Draw  a  diagram  before  starting 
the  solution. 

14.  Draw    a    diagram    simi- 
lar to  the  hands  of  a  clock,  and 
explain    what   is   meant   by   an 
azimuth. 

15.  If  the  limb  of  a  transit 
is  divided  into  20  minute  spaces, 
show  how  the  vernier  must  be 
made  in  order  to  read  1  minute; 

also  how  to  read  20  seconds,     (live  diagrams  of  these  verniers. 

16.  Explain  the  relation  between  azimuth  and  bearings  of  lines 
in  the  four  quadrants.     Determine  the  azimuths  of  the  courses  in 
Problem  3,   page  88. 

17.  Compute  the  area  of  Fig.  A,  taking  the  azimuth  of  BC 
as  0°  00'.     Also  taking  the  azimuth  of  A  B  as  90°  (M)'. 


270 


Ki>r.  A. 


.4  H  =  800  foot. 
H  C  =  500  foot. 
C  I)  =  200  foot. 
I)  K  =  100  foot. 
K  F  =  000  foot. 
FA  =  700  foot. 


A  =  58°  14' 
B  =  120°  00' 
C  =  12o°  00' 
D  =  200°  00' 
E  =  83°  34' 
F  =  133°  12' 


Axs.     Area  =  11  acres  1  rood  1.62  rods. 

18.  Describe  fully  the  various  steps  involved  in  determining 
the  area  of  an  enclosed  field  from  latitudes  and  departures,  and  state 
the  rule  for  calculating  the  area. 

19.  A  polygon  of  six  sides  has  the  following  interior  angles: 


PLANE  SURVEYING 


A  =  58°24';#  -  121°  30';  C  =  127°  45';  E  =  95°19';F  -  133°  2'. 

The  azimuth  AB  is  00°  0'.  Find  the  azimuth  of  each  of  the  other  sides. 

20.     Give  a  complete  description  of  the  surveyor's  compass. 

Explain  fully  how  to  use  the  compass  in  the  field  in  determining  areas. 

21.  Given  the  latitude  of  one 
end   of   a  line  as  +  2,804.4,  its 
longitude  as  +  4,661.3,  its  length 
as  797.2  feet,  and  its  azimuth  as 
115°  44'  28".     Compute  the  lati- 
tude and  longitude  of  the  other 
end.     Draw  a  figure  before  start- 
ing the  solution. 

Ans.  Latitude      =  2,458.2  feet. 
Longitude    =  5,379.4  feet. 

22.  Describe   fully    the    field 
work  of  running  a  traverse. 

23.  Explain    fully    the    field 
work  of  laying  out  angles. 

24.  In  Problem  2,  Page  88, 
the  bearing  of  the  first  course  is 
changed  to  S.  15°  E.     Calculate 


Fig.  JJ. 


the  changed  bearings  of  the  other  courses. 

25.  Describe    fully   how   to    test  and  make  the  adjustments  of 
the  compass  in  their  order. 

26.  Explain  fully  how  to  lay  out  a  true  meridian  from  Polaris. 

27.  Explain  the  application  of, 
and  describe  fully  how  to  carry 
out,  the  "peg"  adjustment  for  the 
transit. 

28.  Define    and    fully   explain 
the    terms:     Daily,  Annual,    and 
Secular  Variation. 

29.  Describe  the  stadia  and  de- 
duce the  formulae  for  its  use. 

30.  How  many  feet    are    rep- 
resented by  one  inch   on  a  scale 
of   TljL)¥?     How  many   acres   are 
represented  by  one  square  inch  on 


PLANE  SURVEYING 


Ans  $  83;1'  fl'et 
S"  I  398.6  acres. 

31.     Compute  the  area  of  Fig.  B,  which  represents  a  recent 
survey  of  a  farm,  from  the  following  data: 

A  B  =  317.8  feet; 

B  F  -  284.3  feet; 

F  A  =  250.5  feet; 

F  C  -  512.7  feet; 

F  D  =  510.0  feet; 
DEF  =  90°  00'; 
E  F  D  =  69°  45'; 
DF  C  =  61°  12'; 
C  FB    =  49°  30'; 

Ans.    Area  =  5  acres  103 
rods  81  sq.  feet. 

32.  I  )escrit>e  fully  the 
field  work  of  carrying  out 
a  stadia  survey  over  un- 
even ground. 

33.  A  triangle  ARC 
lias  sides  with  the  follow- 
ing lengths  and  azimuths : 

A  H,  I  =  312  feet;  /  -  45  degrees. 
H  C,  1  =  540.4  feet;  Z  =  135  degrees. 
C  A,  I  =  624.0 feet;  Z  =  285  degrees. 

Compute  the  latitude  differ- 
ences and  longitude  differences 
and  the  double  longitudes  for 
each  course. 

34.  The  bearing  of  A  C  (Fig. 
C)   is  N  8.J°  E,    that  of  AD 
is  N  46°  E.    Find  the  value  of  the 
angles  C  A  D  and  D  A  E. 

35.  Find    the    angle  ARC 
(Fig.    D)    when  the   bearing  of 
A  R   is  42°  E,  and  that  of  R  C 
is  S  29J°  E.    What  will  R  A   be 
when   the    first    bearing    is    re- 
versed? Fig.  K 


i^EVIEW     QUESTIONS 


THK      S   U  B  J  K  <J  T      OK 


SURVEYIXG 


I'AKT     III 


Station. 

Distance. 

Bearings. 

1 
2 
3 
4 

12.41 

8.25 
4.24 

S21°W 

N83i4°E 

N47°W 

1.  Let  it  be  required  to  prepare  a  table  of  declinations  for 
July  16,  1904,  for  a  point  whose  latitude  is  38°  30',  and  which  lies 
in  the  «  Eastern  Time  "  belt.     The  sun's  apparent  declination  at 
Greenwich  Mean  Noon  for  that  date  is  21°  24.3'  and  the  hourly 
change  is  -  24"  .38. 

2.  The  lengths  and  bearings  of  the  sides  of  a  field  are  aa 
follows : 

It  is  required  to  find 
the  length  of  Course  2  and 
the  bearing  of  Course  3. 

3.  In  your  own  words 
explain  fully  the  various 
methods  of  surveying  areas  by  the  plane-table.  Prepare  a  sketch 
for  each  case. 

4.  Explain    the    term    "  adjusting    the    triangle "   and    why 
adjustment  is  necessary. 

5.  A  certain  grade  line  has  a  fall  of  12.5  feet  in  one-half  of 
a  mile.     Determine  the  percentage  of  grade  and  the  vertical  angle 
corresponding  to  it. 

6.  In    Fig.    103   given   BC  =  385    feet;  ABC  =  70°   05'; 
ACB  =  63°  28'.     Calculate  the  length  of  BD  and  the  length  of 
AD,  and  plot  the  figure  accurately. 

7.  It  is  required  to  determine  the  linear  convergence  for  a 
township  situated  in  latitude  43°  18'  north. 

8.  In  your  own  words  describe  fully  the  adjustments  of  the 
plane-table  in  their  order. 


206  PLANE    SURVEYING 


9.  Explain  under  what  circumstances 'the  transit  and  stadia 
may  be  used  and  when  the  plane-table  may  be  used.  Enumerate 
the  advantages  and  disadvantages  of  each  instrument. 

10.  In  Fig.  95  the  scale  reading  was  3i  and  the  reading  of 
the  head  26.     Determine  the  grade  of  the  line  AB. 

11.  In    Fig.  101  given  HE  =  265  feet;  AHE  =  112°  30'; 
BITE  =  35°  10';  BEH  =  116°;  BEA  =  85°.     It  is  required  to  plot 
the  figure   accurately  and    to   find    the  angle  AHD,    the  length 
of  HD,  and  the  length  of  AD. 

12.  Explain  in  your  own  language  the  methods  of  using  a 
steel  tape  in  the  measurement  of  a  base. 

13.  The  lengths  and  bearings  of  the  sides  of  a  field  are  as 
follows : 

It   is   required  to  find 
the  length   and   bearing  of 


Stations.       Distances. 


1  10.03 

2  4.10 

3  7.  (SO 


Bearings. 

N  52°  E 
S  25)1°  E 


Course  4. 


14.     In    the    measure- 


\  ment  of  a  base  line,  the  tape 


is  divided  into  eight  sections  of  50  feet  each.  The  weight  per  foot 
of  tape  is  .0145  Ibs.  The  tension  applied  to  the  end  of  the  tape  is 
15  Ibs.  Determine  the  amount  of  shortening  of  each  tape  length. 
If  there  are  75  full  tape  lengths  in  measured  base  line,  determine 
the  total  corrections  for  sag. 

15.  Explain  fully  in  your  own  words  the  nature  of  Topo- 
graphical Surveying  and  what  instruments  are  particularly  adapted 
to  this  work. 

1(5.  A  line  is  to  be  run  at  a  grade  of  4.75  j>er  cent.  Explain 
fully  how  this  would  be  done  with  the  gradienter. 

17.  In  Fig.  Ill  given  BE'  =  375  feet;  angle  CBE'  =  112°25(; 
angle  CE'B  =  52"'  16'.     Plot  the  figure  accurately  to  a  scale  of  100 
feet  to  the  inch.     Calculate  the  length  of  BC  and  if  etake  B  is 
numbered  180  +  36,  determine  the  number  of  stake  C. 

18.  In  your  own  words  describe  the  plane-table  and  its  uses; 
its  advantages  and  disadvantages. 

19.  Explain  fully  the  organization    of    a    topographic  party 
using  the  transit  and  stadia,  and  explain  the  method  of  keeping  the 
field -notes. 


INDEX 


The  page  numbers  of  this  volume  will  be  found  at  the  bottom  of  the 
pages;  the  numbers  at  the  top  refer  only  to  the  section. 


Aberration 

Abney  hand-level 

Acute  angle,  definition  of 

Acute-angled  triangle,  definition  of 

Agonic  lines 

Altitude  of  a  star,  definition  of 

Altitude  of  triangle,  definition  of 

Angles,  measurement  of 

Annual  variation 

Assembly  drawing 

Azimuth 

B 

Base  measurement 

apparatus  for 

errors  in 

tape-stretcher 

Base  of  triangle,  definition  of 
Beam  compasses 
Bearings,  to  change 
Bench-mark,  definition  of 
Black  prints,  formula  for 
Blue-print  solution,  formula  for 
Blue  printing 
Boston  rod 
Bow  pen 
Broken  line,  definition  of 


Capital  letters 
Central  angle,  definition  of 
Chaining  on  slopes 
Chord,  definition  of 
Chromatic  aberration 
Circles,  definition  of 


age 

Page 

Clinometer 

39 

55 

Collimation,  line  of 

54 

39 

Compass 

76 

256 

adjustment 

79 

256 

magnetic  needle 

77 

76 

sights 

78 

148 

tangent  scale 

78 

257 

use  of 

80 

260 

Compasses,  drawing 

220 

75 

Cone,  definition  of 

264 

356 

Conic  sections,  definition  of 

205 

97 

Cross-section  rod 

49 

Cross-sectioning 

69 

Cube,  definition  of 

262 

173 

Curved  line,  definition  of 

255 

174 

Cycloid,  definition  of 

267 

174 

Cylinder,  definition  of 

263 

178 

D 

257 

229 

Deflection  angles 

119 

87 

Departures 

88 

66 

Diurnal  variation 

75 

357 

Dividers 

223 

357 

Division  of  land 

99 

355 

Drawing  board 

2  1  3 

49 

Drawing  instruments  and  materials 

211 

224 

beam  compasses 

229 

255 

lx>ard 

213 

bow  pen 

224 

bow  pencil 

224 

232 

compasses 

220 

260 

dividers 

223 

18 

drawing  pen 

224 

259 

erasers 

214 

55 

ink 

22G 

259 

irregular  curve 

228 

Note — Fa 


numbers  see  foot  of  pages. 


II 

INDEX 

Page 

Pa»e 

Drawing  instruments  and  materials 

Geometrical  definitions 

paper 

211 

pyramids 

262 

pencils 

214 

quadrilaterals 

257 

protractor 

227 

sol  ids 

261 

scales 

227 

spheres 

264 

T-square 

215 

surfaces 

256 

thumb  tacks 

213 

triangles 

256 

triangles 

217 

Gothic  letters 

230 

Drawing  paper 

211 

Gradienter 

141 

Drawing  pen 

224 

Gurley  binocular  hand  level 

60 

Dumpy-level 

58 

E 

H 

Ellipse,  definition  of 

265 

Hand-level 

60 

Engineer's  transit 

111 

Abney 

39 

Epicycloid,  definition  of 

268 

Locke 

37 

Equiangular  triangle 

257 

Horizontal  line,  definition  of 

255 

Equilateral  triangle 

256 

Hyperbola,  definition  of 

206 

Hypocycloid,  definition  of 

268 

F 

1 

Farm  surveying 

85 

balancing  t  he  survey 

90 

India  ink 

226 

bearings,  to  change 

87 

Inscril>ed  angle,  definition  of 

260 

calculating  the  content 

91 

Inscriljed  polygon,  defln  tion  of 

260 

field  notes 

86 

Intersecting  lines,  definition  of 

255 

intersections,  method  of 

85 

Intersection  and  development 

311-328 

latitudes  and  departures 

88 

Involute,  definition  of 

268 

progression,  method  of 

85 

Irregular  curve 

228 

proofs  of  accuracy 

86 

Isogonic  lines 

76 

radiation   method  of 

85 

Isometric  projection 

329 

supplying  omissions 

96 

Isosceles  triangle 

257 

Field  notes,  keeping  of 

30 

Field  work  of  measuring  areas 

26 

L 

Frustum  of  a  pyramid,  definition  of 

263 

Land  measure 

14 

Latitude  difference 

88 

G 

Latitude  of  a  place,  deflni'ion  of 

148 

Geodetic  surveying 

11 

Latitudes  and  departures 

88 

Geometrical  definitions 

testing  a  survey  by 

89 

angles 

256 

Lettering 

229.   346 

circles 

259 

capital 

232 

cones 

263 

Gothic 

230 

conic  sections 

265 

lower-case 

232 

cylinders 

263 

Roman 

230 

lines 

255 

\A-\t-\  bubble 

34 

odontoidal  curves 

267 

Leveling 

64 

point 

255 

cross-sectioning 

69 

polygons 

258 

profile 

67 

Note.  —  For  pane  numbers  tee  foot  of 

pages. 

in 

LyiWy. 

III 

Page 

Leveling  instruments 
care  of 

51 

Odontoidal  curves,  definition  of 

Page 

63 

epicycloid 

dumpy-level 
hand  level 

58 
60 

hypocycloid 
involute 

268 
208 

"setting  up" 
spirit  level 
wye  level 

61 
60 
51 

Off-sets  and  tie-lines 
Orthographic  projection 

268 
27 
295 

Leveling  rod 

41 

P 

Line,  definition  of 

255 

Parabola,  definition  of 

266 

Line  of  collimation 

54 

Parallel  lines,  definition  of 

255 

Line  shading 

344 

Parallelogram,  definition  of 

Lines,  measurement  of 

12 

Parallelepiped,  definition  of 

20l' 

Locke's  hand  level 

37 

Peg,  plug,  or  turning  point,  defin 

it  ion  of      oo 

Longitude  difference 

88 

Perpendicular  lines,  definition  of 

255 

Longitude  of  a  point 

88 

Plane  figure,  definition  of 

250 

Lower-case  letters 

232 

Plane  surveying 

1  1  -209 

M 

Plane-table 

182,    201 

PI  us-siglit    definition  of 

06 

Magnetic  declination 

75 

Point,  definition  of 

Magnetic  needle 

77 

Polyedron,  definition  of 

201 

Major  axis 

260 

Polygon,  definition  of 

25s 

Mechanical  drawing 

211-374 

Prism,  definition  of 

261 

blue  printing 

355 

Prismatic  compass 

79 

geometrical  definitions 

255 

Profile  leveling 

67 

geometrical  problems 

269-293 

Profile  plane 

301 

instruments  and  materials 

211 

Project  ion 

intersection  and  development 

311-328 

isometric 

329 

lettering 

229,    346 

oblique 

341 

line  shading 

344 

orthographic 

295 

plates                233-252,   269-293, 

358-374 

third  plane  of 

301 

projections 

295 

Protractor 

227 

shade  lines 

307 

Pyramid,  definition  of 

262 

tracing 

354 

Q 

Meridian 

10'-* 

Meridian  plane 

Quadrilateral,  definition  of 

257 

Minor  axis 

266 

H 

Minus-sight,  definition  of 

66 

Radiating  triangulation 

207 

N 

Radius,  definition  of 

259 

New  York  rod 

47 

Ranging  poles 

50 

Rectangle,  definition  of 

258 

O 

Rectangular  hyperbola 

267 

Obtuse  angle,  definition  of 

256 

Relocation 

82 

Obtuse-angled  triangle 

256 

Resurveys 

97 

Oblique  lines,  definition  of 

255 

Rhomboid,  definition  of 

258 

Oblique  projections 

341 

Rhombus,  definition  of 

258 

Odontoidal  curves,  definition  of 

267 

Right-angled  triangle,  definition  of 

256 

cycloid 

267 

Right  angles,  definition  of 

256 

Note.  —  For  page  numbers  see  foot  of  pages. 

IV 

INDEX 

Page 

Page 

Hod 

Surveying 

Boston 

49 

relocation 

82 

cross-section 

49 

resurveys 

97 

leveling 

41 

stadia 

127 

New  York 

47 

tape 

16 

Roman  capitals 

230 

topographical 

189 

transit 

104 

S 

traversing 

121 

Scales 

227 

trian  Kill  at  ion 

203 

Secant,  definition  of 

259 

vernier 

31 

Sector,  definition  of 

260 

Surveyor's  transit 

111 

Secular  variation 

75 

Shade  lines 

307 

T 

Solar  transit 

149 

T-square 

215 

adjustments  of 

151 

Tables 

use  of 

153 

base  measurement 

173 

Sphere,  definition  of 

204 

field  surveying  data 

96 

Spherical  alxsrration 

55 

land  measure 

14 

Spirit  level 

00 

latitude  coefficients 

156 

Square,  definition  of 

258 

mean  refraction  at  various 

altitudes     155 

Stadia 

127 

I  H.I.  iris  data 

138 

use  of,  in  field 

132 

refraction  correction 

158 

Stadia  rods 

134 

Tachymeter 

111 

Steel  tapes 

175 

Tangent,  definition  of 

259 

Straight  line,  definition  of 

255 

Tape 

10 

Surface,  definition  of 

250 

Tape-stretcher 

178 

Surveying 

11-209 

Telemeter 

134 

azimuth 

»7 

Theodolite 

111 

base  measurement 

173 

Third  plane  of  projection 

301 

Boston  rod 

49 

Thumb  tacks 

213 

clinometer 

39 

Topographical  surveying 

189 

compass 

70 

equipment 

191.    192.   190 

cross-section  rod 

49 

field  operations 

190.    199 

farm 

85 

method  of  pnx-edure 

192 

geodetic 

11 

organization  of  party 

200 

gradienter 

141 

photography 

202 

Gunter's  chain 

13 

plane  table  and  stadia 

201 

hand  level 

37 

transit  and  stadia 

199 

level  bubble 

34 

Tracing 

354 

leveling  instruments 

51 

Transit 

104 

leveling  rod 

41 

adjust  ment 

112 

measurement  of  lines 

12 

engineer's 

11  1 

meridian 

102 

to  "set  up" 

117 

New  York  rrxl 

47 

solar 

149 

plane 

11 

surveyor's 

111 

plane-table 

182 

Transit-theodolite 

111 

ranging  poles 

50 

Trapezium,  definition  of 

257 

Note.  —  For  pni/e  numbers  ace  ft 

lot  of  pages. 

INDEX 


Trapezoid,  definition  of 
Traversing 

checking  the  traverse 

keeping  notes 
Triangles,  definition  of 
Triangulation 

adjusting  triangle 

angles,  measuring 

base  line,  measuring 

radiating 
True  meridian 
Note. — For  page  numbers  see  foot  of  pages. 


Page 

Page 

257 

Truncated  prism,  definition  of 

262 

,    188 

U 

124 

U.  S.  public  land  surveys 

168 

256 

V 

203 

Variation 

75 

209 

Vernier 

31 

205 

Vertical  line,  definition  of 

255 

204 

W 

207 

Wye  level 

51 

102 

adjustments 

55 

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